$\displaystyle=\tilde{a}_{t}+\tilde{\gamma}_{1t}\ln\tilde{\theta}_{t}+\tilde{\gamma}_{2t}\ln\tilde{I}_{t}+\tilde{\gamma}_{3t}\ln\tilde{\theta}_{t}\ln\tilde{I}_{t}+\lambda_{\theta,t+1,1}E[\eta_{\theta,t}\mid\tilde{\eta}_{I,t}]$
$\displaystyle=\tilde{a}_{t}+\tilde{\gamma}_{1t}\ln\tilde{\theta}_{t}+\tilde{\gamma}_{2t}\ln\tilde{I}_{t}+\tilde{\gamma}_{3t}\ln\tilde{\theta}_{t}\ln\tilde{I}_{t}+(\lambda_{\theta,t+1,1}/\lambda_{I,t,1})\kappa_{t}\tilde{\eta}_{I,t}$

The second line follows because $\tilde{\theta}_{t},\tilde{I}_{t},Y_{t}$ is
identical to conditioning on $\tilde{\theta}_{t},\tilde{\eta}_{I,t},Y_{t}$ due
to the line relationship in equation (9). The third and fourth line follow
from Assumption 1(b) and the relationship
$\tilde{\eta}_{I,t}=\lambda_{I,t,1}{\eta}_{I,t}$. Hence, we can identify
$\tilde{a}_{t}$, $\tilde{\gamma}_{1t}$, $\tilde{\gamma}_{2t}$,
$\tilde{\gamma}_{3t}$, and
$(\lambda_{\theta,t+1,1}/\lambda_{I,t,1})\kappa_{t}$ by regressing
$\ln\tilde{\theta}_{t+1}$ on $\ln\tilde{\theta}_{t}$, $\ln\tilde{I}_{t}$,
$\ln\tilde{I}_{t}$, and
$\tilde{\eta}_{I,t}=\ln\tilde{I}_{t}-\tilde{\beta}_{0t}-\tilde{\beta}_{1t}\ln\tilde{\theta}_{t}-\tilde{\beta}_{2t}\ln
Y_{t}$.

Now suppose we have an alternative set of parameters, denoted by
$\\{\bar{\mu}_{\theta,t,m},\bar{\lambda}_{\theta,t,m}\\}_{t=0,\ldots,T,m=1,2}$,
$\\{\bar{\mu}_{I,t,m},\bar{\lambda}_{I,t,m}\\}_{t=0,\ldots,T-1,m=1,2}$,
$\\{\bar{a}_{t},\bar{\gamma}_{1t},\bar{\gamma}_{2t},\bar{\gamma}_{3t}\\}^{T-1}_{t=0}$,
$\\{\bar{\beta}_{0t},\bar{\beta}_{1t},\bar{\beta}_{2t}\\}^{T-1}_{t=0}$, and
$(\bar{\rho}_{0},\bar{\rho}_{1})$ that yields the same values of
$\\{\tilde{\mu}_{\theta,t,m},\tilde{\lambda}_{\theta,t,m}\\}_{t=0,\ldots,T,m=1,2}$,
$\\{\tilde{\mu}_{I,t,m},\tilde{\lambda}_{I,t,m}\\}_{t=0,\ldots,T-1,m=1,2}$,
$\\{\tilde{\beta}_{0t},\tilde{\beta}_{1t},\tilde{\beta}_{2t}\\}^{T-1}_{t=0}$,
$(\tilde{\rho}_{0},\tilde{\rho}_{1})$, and
$\\{\tilde{a}_{t},\tilde{\gamma}_{1t},\tilde{\gamma}_{2t},\tilde{\gamma}_{3t}\\}^{T-1}_{t=0}$.
Define $\bar{\theta}_{t}$ and $\bar{I}_{t}$ such that

$\bar{\mu}_{\theta,t,1}+\bar{\lambda}_{\theta,t,1}\ln\bar{\theta}_{t}={\mu}_{\theta,t,1}+{\lambda}_{\theta,t,1}\ln{\theta}_{t}=\ln\tilde{\theta}_{t}$

and

$\bar{\mu}_{I,t,1}+\bar{\lambda}_{I,t,1}\ln\bar{I}_{t}={\mu}_{I,t,1}+{\lambda}_{I,t,1}\ln{I}_{t}=\ln\tilde{I}_{t}$

For $m\neq 1$, we then have

$\displaystyle\bar{\mu}_{\theta,t,m}+\bar{\lambda}_{\theta,t,m}\ln\bar{\theta}_{t}$
$\displaystyle=\bar{\mu}_{\theta,t,m}+\bar{\lambda}_{\theta,t,m}\left(\ln\tilde{\theta}_{t}-\bar{\mu}_{\theta,t,1}\right)/\bar{\lambda}_{\theta,t,1}$
$\displaystyle=\bar{\mu}_{\theta,t,m}-\frac{\bar{\lambda}_{\theta,t,m}}{\bar{\lambda}_{\theta,t,1}}+\frac{\bar{\lambda}_{\theta,t,m}}{\bar{\lambda}_{\theta,t,1}}\ln\tilde{\theta}_{t}$
$\displaystyle=\tilde{\mu}_{\theta,t,m}+\tilde{\lambda}_{\theta,t,m}\ln\tilde{\theta}_{t}$

and similarly

$\displaystyle\bar{\mu}_{I,t,m}+\bar{\lambda}_{I,t,m}\ln\bar{I}_{t}$
$\displaystyle=\tilde{\mu}_{I,t,m}+\tilde{\lambda}_{I,t,m}\ln\tilde{I}_{t}.$

Then the two models generate the same distribution of the measures. In
addition, the parameters and $\bar{\theta}_{t}$ are consistent with the
production technology because

$\displaystyle\bar{\mu}_{\theta,t+1,1}$
$\displaystyle+\bar{\lambda}_{\theta,t+1,1}\ln\bar{\theta}_{t+1}$
$\displaystyle=\ln\tilde{\theta}_{t+1}$
$\displaystyle=\tilde{a}_{t}+\tilde{\gamma}_{1t}\ln\tilde{\theta}_{t}+\tilde{\gamma}_{2t}\ln
I_{t}+\tilde{\gamma}_{3t}\ln\tilde{\theta}_{t}\ln
I_{t}+\tilde{\eta}_{\theta,t}$
$\displaystyle=\tilde{a}_{t}+\tilde{\gamma}_{1t}(\bar{\mu}_{\theta,t,1}+\bar{\lambda}_{\theta,t,1}\ln\bar{\theta}_{t})+\tilde{\gamma}_{2t}\ln
I_{t}+\tilde{\gamma}_{3t}\ln(\bar{\mu}_{\theta,t,1}+\bar{\lambda}_{\theta,t,1}\ln\bar{\theta}_{t})\ln
I_{t}+\tilde{\eta}_{\theta,t}$
$\displaystyle=\tilde{a}_{t}+\tilde{\gamma}_{1t}(\bar{\mu}_{\theta,t,1}+\bar{\lambda}_{\theta,t,1}\ln\bar{\theta}_{t})+\tilde{\gamma}_{2t}\ln
I_{t}+\tilde{\gamma}_{3t}\ln(\bar{\mu}_{\theta,t,1}+\bar{\lambda}_{\theta,t,1}\ln\bar{\theta}_{t})\ln
I_{t}+\tilde{\eta}_{\theta,t}$

Since

$\tilde{a}_{t}=\bar{\lambda}_{\theta,t+1,1}\bar{a}_{t}+\bar{\mu}_{\theta,t+1,1}-\frac{\bar{\lambda}_{\theta,t+1,1}}{\bar{\lambda}_{\theta,t,1}}\bar{\mu}_{\theta,t,1}\bar{\gamma}_{1t}-\frac{\bar{\lambda}_{\theta,t+1,1}}{\bar{\lambda}_{I,t,1}}\mu_{I,t,1}\bar{\gamma}_{2t}$
$\tilde{\gamma}_{1t}=\bar{\lambda}_{\theta,t+1,1}\left(\frac{1}{\bar{\lambda}_{\theta,t,1}}\bar{\gamma}_{1t}-\frac{\bar{\mu}_{I,t,1}}{\bar{\lambda}_{I,t,1}}\bar{\gamma}_{3t}\right),\;\;\tilde{\gamma}_{2t}=\bar{\lambda}_{\theta,t+1,1}\left(\frac{\bar{\gamma}_{2t}}{\bar{\lambda}_{I,t,1}}-\frac{\bar{\mu}_{\theta,t,1}}{\bar{\lambda}_{\theta,t,1}}\bar{\gamma}_{3t}\right),\;\;\tilde{\gamma}_{3t}=\frac{\bar{\lambda}_{\theta,t+1,1}}{\bar{\lambda}_{\theta,t,1}\bar{\lambda}_{I,t,1}}\bar{\gamma}_{3t}$

and $\tilde{\eta}_{\theta,t}=\lambda_{\theta,t+1,1}{\eta}_{\theta,t}$, it is
easy to show that

$\displaystyle\ln\bar{\theta}_{t+1}$
$\displaystyle=\bar{a}_{t}+\bar{\gamma}_{1t}\ln\bar{\theta}_{t}+\bar{\gamma}_{2t}\ln
I_{t}+\bar{\gamma}_{3t}\ln\bar{\theta}_{t}\ln I_{t}+\bar{\eta}_{\theta,t}$

with
$\bar{\eta}_{\theta,t}=\frac{\lambda_{\theta,t+1,1}}{\bar{\lambda}_{\theta,t+1,1}}{\eta}_{\theta,t}$.

Analogously, one can show that

$\displaystyle\ln\bar{I}_{t}$
$\displaystyle=\bar{\beta}_{0t}+\bar{\beta}_{1t}\ln\theta_{t}+\bar{\beta}_{2t}\ln
Y_{t}+\bar{\eta}_{I,t}$ $\displaystyle Q$
$\displaystyle=\bar{\rho}_{0}+\bar{\rho}_{1}\ln\bar{\theta}_{T}+\bar{\eta}_{Q}$

For the last part first notice that once
$\\{\lambda_{\theta,t,1},\lambda_{I,t,1},\mu_{\theta,t,1},\mu_{I,t,1}\\}_{t=1,\ldots,T}$
are fixed, the vector
$\\{\lambda_{\theta,t,m},\lambda_{I,t,m},\mu_{\theta,t,m},\mu_{I,t,m}\\}_{t=1,\ldots,T}$
is uniquely determined for all $m\neq 1$ due to the restrictions the
parameters in the identified set have to satisfy. From these restrictions, we
can then uniquely determine $\gamma_{3t}$ and with that parameter also
$\gamma_{1t}$ and $\gamma_{2t}$. Similarly, all the other parameters are
uniquely determined. We then found the unique parameter vector for which
$\\{\lambda_{\theta,t,1},\lambda_{I,t,1},\mu_{\theta,t,1},\mu_{I,t,1}\\}_{t=1,\ldots,T}$
are fixed and that yields the same values of
$\\{\tilde{\mu}_{\theta,t,m},\tilde{\lambda}_{\theta,t,m}\\}_{t=0,\ldots,T,m=1,2}$,
$\\{\tilde{\mu}_{I,t,m},\tilde{\lambda}_{I,t,m}\\}_{t=0,\ldots,T-1,m=1,2}$,
$\\{\tilde{a}_{t},\tilde{\gamma}_{1t},\tilde{\gamma}_{2t},\tilde{\gamma}_{3t}\\}^{T-1}_{t=0}$,
$(\tilde{\rho}_{0},\tilde{\rho}_{1})$, and
$\\{\tilde{\beta}_{0t},\tilde{\beta}_{1t},\tilde{\beta}_{2t}\\}^{T-1}_{t=0}$.

∎

###### Proof of Corollary 1.

The last part of Theorem 1 immediately implies that all parameters are point
identified under Assumptions 1, 2, 3(a), and 4(a). Now suppose Assumptions 1,
2, 3(b), and 4(a) hold. Then, since $a_{t}=\mu_{I,t,1}=0$,
$\mu_{\theta,t+1,1}-\mu_{\theta,t,1}\frac{\lambda_{\theta,t+1,1}}{\lambda_{\theta,t,1}}\gamma_{1t}$
is identified. In addition
$\frac{\lambda_{\theta,t+1,1}}{\lambda_{\theta,t,1}}\gamma_{1t}$ is
identified. We also know that $\mu_{\theta,0,1}=0$, which identifies
$\mu_{\theta,1,1}$ and then we can recursively identify $\mu_{\theta,t,1}$ for
all $t$. In addition, we can identify $\lambda_{\theta,t+1,1}\gamma_{2t}$
because
$\lambda_{\theta,t+1,1}\left(\gamma_{2t}-\frac{\mu_{\theta,t,1}}{\lambda_{\theta,t,1}}\gamma_{3t}\right)$,
$\frac{\lambda_{\theta,t+1,1}}{\lambda_{\theta,t,1}}\gamma_{3t}$, and
$\mu_{\theta,t,1}$ are identified. Next notice that, we can also identify

$\frac{\lambda_{\theta,t+1,1}}{\lambda_{\theta,t,1}}\gamma_{1t}=\frac{\lambda_{\theta,t+1,1}}{\lambda_{\theta,t,1}}(1-\gamma_{2t}-\gamma_{3t})=\frac{\lambda_{\theta,t+1,1}}{\lambda_{\theta,t,1}}-\frac{\lambda_{\theta,t+1,1}}{\lambda_{\theta,t,1}}\gamma_{2t}-\frac{\lambda_{\theta,t+1,1}}{\lambda_{\theta,t,1}}\gamma_{3t}$

Since $\lambda_{\theta,0,1}=1$, we can identify $\lambda_{\theta,1,1}$ and
then, using the previous equation, $\lambda_{\theta,t,1}$ recursively for all
$t$. Once we know $\mu_{\theta,t,1}$ and $\lambda_{\theta,t,1}$ for all $t$,
identification of the production function parameters follows immediately. In
addition, since $\frac{\lambda_{\theta,t,m}}{\lambda_{\theta,t,1}}$ and
$\mu_{\theta,t,m}+\left(\lambda_{\theta,t,m}/\lambda_{\theta,t,1}\right)\mu_{\theta,t,1}$
are identified, we can identify $\lambda_{\theta,t,m}$ and $\mu_{\theta,t,m}$
for all $m$. It then also immediately follows that $\beta_{0t}$, $\beta_{1t}$,
$\beta_{2t}$, $\rho_{0}$, and $\rho_{1}$ are identified.

Now suppose Assumptions 1, 2, 3(a), and 4(b) hold. Then, using the argument as
in the proof of Theorem 1,
$\\{\lambda_{\theta,t,m},\mu_{\theta,t,m},\\}_{t=1,\ldots,T}$ is uniquely
determined. Next, using the expressions for $\tilde{\beta}_{1t}$ and
$\tilde{\beta}_{2t}$ and $\beta_{1t}+\beta_{2t}=1$, we can identify
$\beta_{1t}$, $\beta_{2t}$, and $\lambda_{I,t,1}$. With $\tilde{\beta}_{0t}$
and $\beta_{0t}=0$, we can uniquely determine $\mu_{I,t,1}$. Once the
coefficients in the measurement error equation are identified for one of the
measures, the second part of Theorem 1 implies that all other parameters are
identified as well.

Finally suppose Assumptions 1, 2, 3(b), and 4(b) hold. Then, using
$\beta_{0t}=0$ and $\beta_{1t}+\beta_{2t}=1$ we can write
$\tilde{\beta}_{0t}=\mu_{I,t,1}-\tilde{\beta}_{1t}\mu_{\theta,t,1}$ and
$\lambda_{\theta,t,1}\tilde{\beta}_{1t}+\tilde{\beta}_{2t}=\lambda_{I,t,1}$.
Now assume that $\mu_{\theta,t,1}$ and $\lambda_{\theta,t,1}$, are identified.
Then $\mu_{I,t,1}$ and $\lambda_{I,t,1}$ are also identified. Notice that
since $\gamma_{1t}+\gamma_{2t}+\gamma_{2t}=1$, we have

$\frac{\tilde{\gamma}_{1t}}{\lambda_{I,t,1}}+\frac{\tilde{\gamma}_{2t}}{\lambda_{\theta,t,1}}+\tilde{\gamma}_{3t}=\frac{\lambda_{\theta,t+1,1}}{{\lambda_{\theta,t,1}\lambda_{I,t,1}}}-\mu_{I,t,1}\frac{\lambda_{\theta,t,1}}{\lambda_{I,t,1}}\tilde{\gamma}_{3t}-\mu_{\theta,t,1}\frac{\lambda_{I,t,1}}{\lambda_{\theta,t,1}}\tilde{\gamma}_{3t}$

and thus, $\lambda_{\theta,t+1,1}$ and
$\lambda_{I,t+1,1}=\lambda_{\theta,t+1,1}\tilde{\beta}_{1,t+1}+\tilde{\beta}_{2,t+1}$
are identified. Then $\gamma_{1t}$, $\gamma_{2t}$, and $\gamma_{3t}$ are also
identified. Using the expression for $\tilde{a}_{t}$ together with $a_{t}=0$
then identifies $\mu_{\theta,t+1,1}$ and
$\mu_{I,t+1,1}=\tilde{\beta}_{0,t+1}+\tilde{\beta}_{1,t+1}\mu_{\theta,t+1,1}$.
Since $\mu_{\theta,0,1}=0$ and $\lambda_{\theta,0,1}=1$, these arguments imply
that $\mu_{\theta,t,1}$, $\mu_{I,t,1}$, $\lambda_{\theta,t,1}$, and
$\lambda_{I,t,1}$, are identified for all $t$. Once the coefficients in the
measurement error equation are identified for one of the measures, the second
part of Theorem 1 implies that all other parameters are identified as well.

∎

###### Proof of Theorem 2.

For the first part, notice that

$\displaystyle
F_{\ln\tilde{\theta}_{t+1}}(\tilde{a}_{t}+\tilde{\gamma}_{1t}\ln
Q_{\alpha_{1}}(\tilde{\theta}_{t})+\tilde{\gamma}_{2t}\ln
Q_{\alpha_{2}}(\tilde{I}_{t})+\tilde{\gamma}_{3t}\ln
Q_{\alpha_{1}}(\tilde{\theta}_{t})\ln
Q_{\alpha_{2}}(\tilde{I}_{t})+Q_{\alpha_{3}}(\tilde{\eta}_{\theta,t}))$
$\displaystyle=$ $\displaystyle
F_{\ln\tilde{\theta}_{t+1}}\left(\tilde{a}_{t}+\tilde{\gamma}_{1t}Q_{\alpha_{1}}(\mu_{\theta,t,1}+\lambda_{\theta,t,1}\ln\theta_{t})+\tilde{\gamma}_{2t}Q_{\alpha_{2}}(\mu_{I,t,1}+\lambda_{I,t,1}\ln
I_{t})\right.$
$\displaystyle\qquad\qquad+\left.\tilde{\gamma}_{3t}Q_{\alpha_{1}}(\mu_{\theta,t,1}+\lambda_{\theta,t,1}\ln\theta_{t})Q_{\alpha_{2}}(\mu_{I,t,1}+\lambda_{I,t,1}\ln
I_{t})+Q_{\alpha_{3}}(\lambda_{\theta,t+1,1}\eta_{\theta,t})\right)$
$\displaystyle=$ $\displaystyle
F_{\ln\tilde{\theta}_{t+1}}\left(\tilde{a}_{t}+\tilde{\gamma}_{1t}(\mu_{\theta,t,1}+\lambda_{\theta,t,1}Q_{\alpha_{1}}\ln(\theta_{t}))+\tilde{\gamma}_{2t}(\mu_{I,t,1}+\lambda_{I,t,1}Q_{\alpha_{2}}(\ln
I_{t}))\right.$
$\displaystyle\qquad\qquad+\left.\tilde{\gamma}_{3t}(\mu_{\theta,t,1}+\lambda_{\theta,t,1}Q_{\alpha_{1}}\ln(\theta_{t}))(\mu_{I,t,1}+\lambda_{I,t,1}Q_{\alpha_{2}}(\ln
I_{t}))+\lambda_{\theta,t+1,1}Q_{\alpha_{3}}(\eta_{\theta,t})\right)$
$\displaystyle=$ $\displaystyle
F_{\ln\tilde{\theta}_{t+1}}\left(\mu_{t+1}+\lambda_{t+1}\left({a}_{t}+{\gamma}_{1t}Q_{\alpha_{1}}\ln(\theta_{t})+{\gamma}_{2t}Q_{\alpha_{2}}(\ln
I_{t})+{\gamma}_{3t}Q_{\alpha_{1}}\ln(\theta_{t})Q_{\alpha_{2}}(\ln
I_{t})+Q_{\alpha_{3}}(\eta_{\theta,t})\right)\right)$ $\displaystyle=$
$\displaystyle
F_{\ln{\theta}_{t+1}}\left({a}_{t}+{\gamma}_{1t}Q_{\alpha_{1}}\ln(\theta_{t})+{\gamma}_{2t}Q_{\alpha_{2}}(\ln
I_{t})+{\gamma}_{3t}Q_{\alpha_{1}}\ln(\theta_{t})Q_{\alpha_{2}}(\ln
I_{t})+Q_{\alpha_{3}}(\eta_{\theta,t})\right)$

Since the joint distribution of
$\left(\\{\ln\tilde{\theta}_{t}\\}_{t=0,\dots,T},\\{\ln\tilde{I}_{t}\\}_{t=0,\dots,T-1}\right)$
and $\tilde{a}_{t}$, $\tilde{\gamma}_{1t}$, $\tilde{\gamma}_{2t}$, and
$\tilde{\gamma}_{3t}$ are identified and since
$\tilde{\eta}_{\theta,t}=\ln\tilde{\theta}_{t+1}-\left(\tilde{a}_{t}+\tilde{\gamma}_{1t}\ln\tilde{\theta}_{t}+\tilde{\gamma}_{2t}\ln
I_{t}+\tilde{\gamma}_{3t}\ln\tilde{\theta}_{t}\ln I_{t}\right)$, it follows
that

$F_{\ln\tilde{\theta}_{t+1}}(\tilde{a}_{t}+\tilde{\gamma}_{1t}\ln
Q_{\alpha_{1}}(\tilde{\theta}_{t})+\tilde{\gamma}_{2t}\ln
Q_{\alpha_{2}}(\tilde{I}_{t})+\tilde{\gamma}_{3t}\ln
Q_{\alpha_{1}}(\tilde{\theta}_{t})\ln
Q_{\alpha_{2}}(\tilde{I}_{t})+Q_{\alpha_{3}}(\tilde{\eta}_{\theta,t}))$

and therefore

$F_{\ln{\theta}_{t+1}}\left({a}_{t}+{\gamma}_{1t}Q_{\alpha_{1}}\ln(\theta_{t})+{\gamma}_{2t}Q_{\alpha_{2}}(\ln
I_{t})+{\gamma}_{3t}Q_{\alpha_{1}}\ln(\theta_{t})Q_{\alpha_{2}}(\ln
I_{t})+Q_{\alpha_{3}}(\eta_{\theta,t})\right)$

is point identified.

For the second part notice that

$\displaystyle\ln\tilde{I}_{t}(Y)$
$\displaystyle=\tilde{\beta}_{0t}+\tilde{\beta}_{1t}\ln
Q_{\alpha_{1}}(\tilde{\theta}_{t})+\tilde{\beta}_{2t}\ln
Y+Q_{\alpha_{4}}(\tilde{\eta}_{I,t})$
$\displaystyle=\tilde{\beta}_{0t}+\tilde{\beta}_{1t}(\mu_{\theta,t,1}+\lambda_{\theta,t,1}Q_{\alpha_{1}}\ln(\theta_{t}))+\tilde{\beta}_{2t}\ln
Y+Q_{\alpha_{4}}(\lambda_{I,t,1}{\eta}_{I,t})$
$\displaystyle=\mu_{I,t,1}+\lambda_{I,t,1}\left(\beta_{0t}+\beta_{1t}Q_{\alpha_{1}}(\ln(\theta_{t}))+{\beta}_{2t}\ln
Y+Q_{\alpha_{4}}({\eta}_{I,t})\right)$
$\displaystyle=\mu_{I,t,1}+\lambda_{I,t,1}\ln I_{t}(Y)$

is identified. Then using the same arguments as above, it follows

$\displaystyle
F_{\ln\tilde{\theta}_{t+1}}(\tilde{a}_{t}+\tilde{\gamma}_{1t}\ln
Q_{\alpha_{1}}(\tilde{\theta}_{t})+\tilde{\gamma}_{2t}\ln\tilde{I}_{t}(Y)+\tilde{\gamma}_{3t}\ln
Q_{\alpha_{1}}(\tilde{\theta}_{t})\ln\tilde{I}_{t}(Y)+Q_{\alpha_{3}}(\tilde{\eta}_{\theta,t}))$
$\displaystyle=F_{\ln\theta_{t+1}}(a_{t}+\gamma_{1t}\ln
Q_{\alpha_{1}}(\theta_{t})+\gamma_{2t}\ln I_{t}(Y)+\gamma_{3t}\ln
Q_{\alpha_{1}}(\theta_{t})\ln I_{t}(Y)+Q_{\alpha_{3}}(\eta_{\theta,t}))$

is point identified for all $\alpha_{1},\alpha_{3},\alpha_{4}\in(0,1)$.

For the third part we have

$\displaystyle P\left(Q\leq
q\mid\theta_{t}=Q_{\alpha_{1}}(\theta_{t}),\\{I_{s}=Q_{\alpha_{2s}}(I_{s})\\}^{T-1}_{s=t},\\{\eta_{\theta,s}=Q_{\alpha_{3s}}(\eta_{\theta,s})\\}^{T-1}_{s=t}\right)$
$\displaystyle=P\left(Q\leq
q\mid\tilde{\theta}_{t}=Q_{\alpha_{1}}(\tilde{\theta}_{t}),\\{\tilde{I}_{s}=Q_{\alpha_{2s}}(\tilde{I}_{s})\\}^{T-1}_{s=t},\\{\tilde{\eta}_{\theta,s}=Q_{\alpha_{3s}}(\tilde{\eta}_{\theta,s})\\}^{T-1}_{s=t}\right)$

and hence, the left hand is identified.

The fourth part follows from identification of the joint distribution of
$(Q,\tilde{\theta}_{t},\\{Y_{s}\\}^{T-1}_{s=t})$

Finally, notice that

$\tilde{\gamma}_{1t}+\tilde{\gamma}_{3t}\ln\tilde{I}_{t}=\frac{\lambda_{\theta,t+1,1}}{\lambda_{\theta,t,1}}\left(\gamma_{1t}+\gamma_{3t}\ln
I_{t}\right)$

whose distribution is identified under Assumption 3(a).

∎

###### Proof of Theorem 3 .

The same arguments as in the proof of Theorem 1 implies that
$\tilde{\beta}_{0t}$, $\tilde{\beta}_{1t}$, $\tilde{\beta}_{2t}$,
$\tilde{\rho}_{0}$ and $\tilde{\rho}_{1}$ are identified. Moreover, we can
also identify
$E[\ln\tilde{\theta}_{t+1}\mid\tilde{\theta}_{t},\tilde{I}_{t},Y_{t}]$ which
we can write as

$\displaystyle
E[\ln\tilde{\theta}_{t+1}\mid\tilde{\theta}_{t},\tilde{I}_{t},Y_{t}]$
$\displaystyle=\frac{\lambda_{\theta,t+1,1}\psi_{t}}{{\sigma}_{t}}\ln\left(\tilde{\gamma}_{1t}\tilde{\theta}_{t}^{\frac{{\sigma}_{t}}{\lambda_{\theta,t,1}}}+\tilde{\gamma}_{2t}\tilde{I}_{t}^{\frac{{\sigma}_{t}}{\lambda_{I,t,1}}}\right)+E[\tilde{\eta}_{\theta,t}\mid\tilde{\theta}_{t},\tilde{I}_{t},Y_{t}]$
$\displaystyle=\frac{\lambda_{\theta,t+1,1}\psi_{t}}{{\sigma}_{t}}\ln\left(\tilde{\gamma}_{1t}\tilde{\theta}_{t}^{\frac{{\sigma}_{t}}{\lambda_{\theta,t,1}}}+\tilde{\gamma}_{2t}\tilde{I}_{t}^{\frac{{\sigma}_{t}}{\lambda_{I,t,1}}}\right)+(\lambda_{\theta,t+1,1}/\lambda_{I,t,1})\kappa_{t}\tilde{\eta}_{I,t}$

Hence

$\displaystyle\frac{\frac{\partial
E[\ln\tilde{\theta}_{t+1}\mid\tilde{\theta}_{t},\tilde{I}_{t},Y_{t}]}{\partial\tilde{\theta}_{t}}}{\frac{\partial
E[\ln\tilde{\theta}_{t+1}\mid\tilde{\theta}_{t},\tilde{I}_{t},Y_{t}]}{\partial\tilde{I}_{t}}}$
$\displaystyle=\frac{\lambda_{I,t,1}}{\lambda_{\theta,t,1}}\frac{\tilde{\gamma}_{1t}}{\tilde{\gamma}_{2t}}\tilde{\theta}_{t}^{\frac{{\sigma}_{t}}{\lambda_{\theta,t,1}}-1}\tilde{I}_{t}^{1-\frac{{\sigma}_{t}}{\lambda_{I,t,1}}}$

which identifies
$\frac{\lambda_{I,t,1}}{\lambda_{\theta,t,1}}\frac{\tilde{\gamma}_{1t}}{\tilde{\gamma}_{2t}}$,
$\frac{{\sigma}_{t}}{\lambda_{\theta,t,1}}$ and
$\frac{{\sigma}_{t}}{\lambda_{I,t,1}}$. Hence,
$\frac{\tilde{\gamma}_{1t}}{\tilde{\gamma}_{2t}}$ is identified. In addition
$(\lambda_{\theta,t+1,1}/\lambda_{I,t,1})\kappa_{t}$ is identified. Now write

$\displaystyle
E[\ln\tilde{\theta}_{t+1}\mid\tilde{\theta}_{t},\tilde{I}_{t},Y_{t}]$
$\displaystyle=\frac{\lambda_{\theta,t+1,1}\psi_{t}}{{\sigma}_{t}}\ln\left(\frac{\tilde{\gamma}_{1t}}{\tilde{\gamma}_{2t}}\tilde{\theta}_{t}^{\frac{{\sigma}_{t}}{\lambda_{\theta,t,1}}\psi_{t}}+\tilde{I}_{t}^{\frac{{\sigma}_{t}}{\lambda_{I,t,1}}}\right)+\frac{\lambda_{\theta,t+1,1}\psi_{t}}{{\sigma}_{t}}\ln\tilde{\gamma}_{2t}+(\lambda_{\theta,t+1,1}/\lambda_{I,t,1})\kappa_{t}\tilde{\eta}_{I,t}$

which is linear in
$\ln\left(\frac{\tilde{\gamma}_{1t}}{\tilde{\gamma}_{2t}}\tilde{\theta}_{t}^{\frac{{\sigma}_{t}}{\lambda_{\theta,t,1}}}+\tilde{I}_{t}^{\frac{{\sigma}_{t}}{\lambda_{I,t,1}}}\right)$
and therefore $\frac{\lambda_{\theta,t+1,1}\psi_{t}}{{\sigma}_{t}}$ and
$\frac{\lambda_{\theta,t+1,1}\psi_{t}}{{\sigma}_{t}}\ln\tilde{\gamma}_{2t}$
are identified. Hence $\tilde{\gamma}_{1t}$ and $\tilde{\gamma}_{2t}$ are
identified.

Now suppose we have an alternative set of parameters, denoted by
$\\{\bar{\mu}_{\theta,t,m},\bar{\lambda}_{\theta,t,m}\\}_{t=0,\ldots,T,m=1,2}$,
$\\{\bar{\mu}_{I,t,m},\bar{\lambda}_{I,t,m}\\}_{t=0,\ldots,T-1,m=1,2}$,
$\\{\bar{a}_{t},\bar{\gamma}_{1t},\bar{\gamma}_{2t},\bar{\gamma}_{3t},\bar{\psi}_{t}\\}^{T-1}_{t=0}$,
$\\{\bar{\beta}_{0t},\bar{\beta}_{1t},\bar{\beta}_{2t}\\}^{T-1}_{t=0}$, and
$(\bar{\rho}_{0},\bar{\rho}_{1})$ that yields the same values of
$\\{\tilde{\mu}_{\theta,t,m},\tilde{\lambda}_{\theta,t,m}\\}_{t=0,\ldots,T,m=1,2}$,
$\\{\tilde{\mu}_{I,t,m},\tilde{\lambda}_{I,t,m}\\}_{t=0,\ldots,T-1,m=1,2}$,
$\\{\tilde{\beta}_{0t},\tilde{\beta}_{1t},\tilde{\beta}_{2t}\\}^{T-1}_{t=0}$,
$\\{\tilde{\gamma}_{1t},\tilde{\gamma}_{2t},\frac{{\sigma}_{t}}{\lambda_{\theta,t,1}},\frac{{\sigma}_{t}}{\lambda_{I,t,1}},\frac{\lambda_{\theta,t+1,1}\psi_{t}}{{\sigma}_{t}}\\}^{T-1}_{t=0}$,
and $(\tilde{\rho}_{0},\tilde{\rho}_{1})$. Define $\bar{\theta}_{t}$ and
$\bar{I}_{t}$ such that

$\bar{\mu}_{\theta,t,1}+\bar{\lambda}_{\theta,t,1}\ln\bar{\theta}_{t}={\mu}_{\theta,t,1}+{\lambda}_{\theta,t,1}\ln{\theta}_{t}=\ln\tilde{\theta}_{t}$

and

$\bar{\mu}_{I,t,1}+\bar{\lambda}_{I,t,1}\ln\bar{I}_{t}={\mu}_{I,t,1}+{\lambda}_{I,t,1}\ln{I}_{t}=\ln\tilde{I}_{t}$

For $m\neq 1$, the arguments of the proof of Theorem 1 imply that

$\displaystyle\bar{\mu}_{\theta,t,m}+\bar{\lambda}_{\theta,t,m}\ln\bar{\theta}_{t}$
$\displaystyle=\tilde{\mu}_{\theta,t,m}+\tilde{\lambda}_{\theta,t,m}\ln\tilde{\theta}_{t}$

and

$\displaystyle\bar{\mu}_{I,t,m}+\bar{\lambda}_{I,t,m}\ln\bar{I}_{t}$
$\displaystyle=\tilde{\mu}_{I,t,m}+\tilde{\lambda}_{I,t,m}\ln\tilde{I}_{t}$

Then the two models generate the same distribution of the measures. In
addition, the parameters and $\bar{\theta}_{t}$ are consistent with the
production technology because

$\displaystyle\exp(\bar{\mu}_{\theta,t+1,1})\bar{\theta}_{t+1}^{\bar{\lambda}_{\theta,t+1,1}}$
$\displaystyle=\tilde{\theta}_{t+1}$
$\displaystyle=\left(\tilde{\gamma}_{1t}\tilde{\theta}_{t}^{\frac{{\sigma}_{t}}{\lambda_{\theta,t,1}}}+\tilde{\gamma}_{2t}\tilde{I}_{t}^{\frac{{\sigma}_{t}}{\lambda_{I,t,1}}}\right)^{\frac{\lambda_{\theta,t+1,1}\psi_{t}}{{\sigma}_{t}}}\exp(\tilde{\eta}_{\theta,t})$
$\displaystyle=\left(\tilde{\gamma}_{1t}\exp\left(\sigma_{t}\frac{\bar{\mu}_{\theta,t,1}}{\lambda_{\theta,t,1}}\right)\bar{\theta}_{t}^{\frac{\bar{\lambda}_{\theta,t,1}}{\lambda_{\theta,t,1}}{\sigma}_{t}}+\tilde{\gamma}_{2t}\exp\left(\sigma_{t}\frac{\bar{\mu}_{I,t,1}}{\lambda_{I,t,1}}\right)\bar{I}_{t}^{\frac{\bar{\lambda}_{I,t,1}}{\lambda_{I,t,1}}{\sigma}_{t}}\right)^{\frac{\lambda_{\theta,t+1,1}\psi_{t}}{{\sigma}_{t}}}\exp(\tilde{\eta}_{\theta,t})$

and thus

$\displaystyle\bar{\theta}_{t+1}$
$\displaystyle=\exp\left(-\frac{\bar{\mu}_{\theta,t+1,1}}{\bar{\lambda}_{\theta,t+1,1}}\right)\left(\tilde{\gamma}_{1t}\exp\left(\sigma_{t}\frac{\bar{\mu}_{\theta,t,1}}{\lambda_{\theta,t,1}}\right)\bar{\theta}_{t}^{\frac{\bar{\lambda}_{\theta,t,1}}{\lambda_{\theta,t,1}}{\sigma}_{t}}+\tilde{\gamma}_{2t}\exp\left(\sigma_{t}\frac{\bar{\mu}_{I,t,1}}{\lambda_{I,t,1}}\right)\bar{I}_{t}^{\frac{\bar{\lambda}_{I,t,1}}{\lambda_{I,t,1}}{\sigma}_{t}}\right)^{\frac{\lambda_{\theta,t+1,1}\psi_{t}}{{\bar{\lambda}_{\theta,t+1,1}}{\sigma}_{t}}}\exp(\tilde{\eta}_{\theta,t}/{\bar{\lambda}_{\theta,t+1,1}})$

Since
$\bar{\lambda}_{\theta,t,1}/\bar{\sigma}_{t}=\lambda_{\theta,t,1}/\sigma_{t}$,
$\bar{\lambda}_{I,t,1}/\bar{\sigma}_{t}=\lambda_{I,t,1}/\sigma_{t}$ and
$\bar{\lambda}_{\theta,t+1,1}\bar{\psi}_{t}/\bar{\sigma}_{t}=\lambda_{\theta,t+1,1}\psi_{t}/\sigma_{t}$
we get

$\displaystyle\bar{\theta}_{t+1}$
$\displaystyle=\left(\tilde{\gamma}_{1t}\exp\left(\bar{\sigma}_{t}\left(\frac{\bar{\mu}_{\theta,t,1}}{\bar{\lambda}_{\theta,t,1}}-\frac{\bar{\mu}_{\theta,t+1,1}}{\bar{\lambda}_{\theta,t+1,1}}\right)\right)\bar{\theta}_{t}^{\bar{\sigma}_{t}}+\tilde{\gamma}_{2t}\exp\left(\bar{\sigma}_{t}\left(\frac{\bar{\mu}_{I,t,1}}{\bar{\lambda}_{I,t,1}}-\frac{\bar{\mu}_{\theta,t+1,1}}{\bar{\lambda}_{\theta,t+1,1}}\right)\right)\bar{I}_{t}^{\bar{\sigma}_{t}}\right)^{\frac{\bar{\psi}_{t}}{\bar{\sigma}_{t}}}\exp(\tilde{\eta}_{\theta,t}/{\bar{\lambda}_{\theta,t+1,1}})$
$\displaystyle=\left(\bar{\gamma}_{1t}\bar{\theta}_{t}^{\bar{\sigma}_{t}}+\bar{\gamma}_{2t}\bar{I}_{t}^{\bar{\sigma}_{t}}\right)^{\frac{\bar{\psi}_{t}}{\bar{\sigma}_{t}}}\exp(\bar{\eta}_{\theta,t})$

with
$\bar{\eta}_{\theta,t}=\frac{\lambda_{\theta,t+1,1}}{\bar{\lambda}_{\theta,t+1,1}}{\eta}_{\theta,t}$.

Analogously, one can show that

$\displaystyle\ln\bar{I}_{t}$
$\displaystyle=\bar{\beta}_{0t}+\bar{\beta}_{1t}\ln\theta_{t}+\bar{\beta}_{2t}\ln
Y_{t}+\bar{\eta}_{I,t}$ $\displaystyle Q$
$\displaystyle=\bar{\rho}_{0}+\bar{\rho}_{1}\ln\bar{\theta}_{T}+\bar{\eta}_{Q}$

For the second part, notice that identification of
$\frac{{\sigma}_{t}}{\lambda_{\theta,t,1}}$,
$\frac{{\sigma}_{t}}{\lambda_{I,t,1}}$, and
$\frac{{\sigma}_{t}}{\lambda_{\theta,t+1,1}\psi_{t}}$ implies identification
of $\frac{\lambda_{I,t,1}}{\lambda_{\theta,t,1}}$. Hence, once
$\lambda_{\theta,t,1}$ is fixed, we can identify $\lambda_{I,t,1}$ and
$\sigma_{t}$ for all $t=1,\ldots,T-1$ and then also $\psi_{t}$. Using the
expression for $\tilde{\gamma}_{1t}$ and $\tilde{\gamma}_{2t}$, it is easy to
see that ${\gamma}_{1t}$ and ${\gamma}_{2t}$ are identified, once
$\mu_{\theta,t,1}$ and $\mu_{\theta,t,2}$ are fixed for all $t$.
Identification of the remaining parameters follows from arguments as those in
the proof of Theorem 1

∎

###### Proof of Corollary 2 .

The first part of Theorem 3 implies that can identify $\sigma_{t}$,
$\lambda_{\theta,t,1}$, $\lambda_{I,t,1}$, and $\psi_{t}$ for all $t$ under
Assumptions 1 and 2’ and either Assumption 5’(a) or Assumption 5’(b).

Now suppose that in addition Assumptions 3’(a) and 4’(a) hold. The last part
of Theorem 3 immediately imply that all parameters are then point identified.

Next suppose that in addition to Assumptions 1, 2’, and either 5’(a) or 5’(b)
Assumptions 3(a) and 4(b) hold. Then $\\{\mu_{\theta,t,m},\\}_{t=1,\ldots,T}$
is uniquely determined. Moreover, we have
$\tilde{\beta}_{0t}=\mu_{I,t,1}-\tilde{\beta}_{1t}\mu_{\theta,t,1}$, implying
that $\\{\mu_{I,t,m},\\}_{t=1,\ldots,T-1}$ is identified. Once the
coefficients in the measurement error equation are identified for one of the
measures, the second part of Theorem 3 implies that all other parameters are
identified as well.

Next suppose that in addition to Assumptions 1, 2’, and either 5’(a) or 5’(b)
Assumptions3’(b) and 4’(a) hold. Notice that

$\tilde{\gamma}_{1t}=\gamma_{1t}\exp\left(\sigma_{t}\left(\frac{\mu_{\theta,t+1,1}}{\lambda_{\theta,t+1,1}}-\frac{\mu_{\theta,t,1}}{\lambda_{\theta,t,1}}\right)\right)\quad\text{and}\quad\tilde{\gamma}_{2t}=\gamma_{2t}\exp\left(\sigma_{t}\left(\frac{\mu_{\theta,t+1,1}}{\lambda_{\theta,t+1,1}}\right)\right)$

and

$\displaystyle 1$
$\displaystyle=\gamma_{1t}+\gamma_{2t}=\exp\left(-\sigma_{t}\frac{\mu_{\theta,t+1,1}}{\lambda_{\theta,t+1,1}}\right)\left(\exp\left(\sigma_{t}\frac{\mu_{\theta,t,1}}{\lambda_{\theta,t,1}}\right)\tilde{\gamma}_{1t}+\tilde{\gamma}_{2t}\right)$

Since $\mu_{\theta,0,1}=0$ and the right hand side is strictly monotone in
$\mu_{\theta,1,1}$, we can identify $\mu_{\theta,1,1}$ and then recursively
$\mu_{\theta,t,1}$ for all $t$. Once the coefficients in the measurement error
equation are identified for one of the measures, the second part of Theorem 3
implies that all other parameters are identified as well.

Finally, suppose that in addition to Assumptions 1, 2’, and either 5’(a) or
5’(b) Assumptions 3’(b) and 4’(b) hold. Notice that

$\tilde{\beta}_{0t}=\mu_{I,t,1}-\tilde{\beta}_{1t}\mu_{\theta,t,1}$

and

$\displaystyle
1=\exp\left(-\sigma_{t}\frac{\mu_{\theta,t+1,1}}{\lambda_{\theta,t+1,1}}\right)\left(\exp\left(\sigma_{t}\frac{\mu_{\theta,t,1}}{\lambda_{\theta,t,1}}\right)\tilde{\gamma}_{1t}+\exp\left(\sigma_{t}\frac{\mu_{I,t,1}}{\lambda_{I,t,1}}\right)\tilde{\gamma}_{2t}\right)$

Using $\mu_{\theta,0,1}=0$ and the first equation, we can determine
$\mu_{0,I,1}$. Given $\mu_{\theta,0,1}=0$ and $\mu_{0,I,1}$ and the second
equation, we can determine $\mu_{\theta,1,1}$. Then using recursion, we can
identify $\mu_{\theta,t,1}$ and $\mu_{I,t,1}$ for all $t$. Once the
coefficients in the measurement error equation are identified for one of the
measures, the second part of Theorem 3 implies that all other parameters are
identified as well.

∎

###### Proof of Theorem 4.

The first four parts proof is analogous to the proof of Theorem 2. For the
last part notice that can identify
$\lambda_{\theta,t+1,1}/\lambda_{\theta,t,1}$ and
$\lambda_{\theta,t,1}/\lambda_{I,t,1}$ for all $t$ under Assumption 1 and
either Assumption 5’(a) or Assumption 5’(b). Moreover,

$\frac{\partial\ln\theta_{t+1}}{\partial\ln\theta_{t}}=\frac{{\lambda}_{\theta,t,1}}{{\lambda}_{\theta,t+1,1}}\frac{\partial\ln\tilde{\theta}_{t+1}}{\partial\ln\tilde{\theta}_{t}}=\frac{{\lambda}_{\theta,t,1}}{{\lambda}_{\theta,t+1,1}}\frac{\partial}{\partial\ln\tilde{\theta}_{t}}\left(\tilde{\gamma}_{1t}\tilde{\theta}_{t}^{\frac{\sigma_{t}}{\lambda_{\theta,t,1}}}+\tilde{\gamma}_{2t}\tilde{I}_{t}^{\frac{\sigma_{t}}{\lambda_{I,t,1}}}\right)^{\frac{\lambda_{\theta,t+1,1}\psi_{t}}{\sigma_{t}}}$

and

$\frac{\partial\ln\theta_{t+1}}{\partial\ln
I_{t}}=\frac{{\lambda}_{I,t,1}}{{\lambda}_{\theta,t+1,1}}\frac{\partial\ln\tilde{\theta}_{t+1}}{\partial\ln\tilde{I}_{t}}=\frac{{\lambda}_{I,t,1}}{{\lambda}_{\theta,t,1}}\frac{{\lambda}_{\theta,t,1}}{{\lambda}_{\theta,t+1,1}}\frac{\partial}{\partial\ln\tilde{I}_{t}}\left(\tilde{\gamma}_{1t}\tilde{\theta}_{t}^{\frac{\sigma_{t}}{\lambda_{\theta,t,1}}}+\tilde{\gamma}_{2t}\tilde{I}_{t}^{\frac{\sigma_{t}}{\lambda_{I,t,1}}}\right)^{\frac{\lambda_{\theta,t+1,1}\psi_{t}}{\sigma_{t}}}$

whose distributions are then identified. ∎

###### Proof of Theorem 5.

The joint distribution of

$\\{\\{Z_{\theta,t,m}\\}_{t=0,\ldots,T,m=1,2,3},\\{Z_{I,t,m}\\}_{t=0,\ldots,T-1,m=1,2,3},Q,\\{\tilde{g}_{\theta,t}(\theta_{t})\\}^{T}_{t=0},\\{\tilde{g}_{I,t}(I_{t})\\}^{T-1}_{t=0}\\}$

is identified by Theorem 1 of CHS:10 up to unknown and strictly increasing
functions $\tilde{g}_{\theta,t}$ and $\tilde{g}_{I,t}$ conditional on
$\\{Y_{1},\ldots,Y_{T-1}\\}$. Moreover, let

$\tilde{\eta}_{I,t}=F_{\eta_{I,t}}(\eta_{I,t})\sim U[0,1]$

Then we can write

$\displaystyle I_{t}$
$\displaystyle=h_{t}(\theta_{t},Y_{t},F^{-1}_{\eta_{I,t}}(\tilde{\eta}_{I,t}))$

and thus,

$\displaystyle\tilde{g}_{I,t}(I_{t})$
$\displaystyle=\tilde{g}_{I,t}(h_{t}(\tilde{g}^{-1}(\tilde{g}_{\theta,t}(\theta_{t})),Y_{t},F^{-1}_{\eta_{I,t}}(\tilde{\eta}_{I,t})))$

or

$\displaystyle\tilde{g}_{I,t}(I_{t})$
$\displaystyle=\tilde{h}_{t}(\tilde{g}_{\theta,t}(\theta_{t}),Y_{t},\tilde{\eta}_{I,t})$

Now notice that

$Q_{\alpha}(\tilde{g}_{I,t}(I_{t})\mid\tilde{g}_{\theta,t}(\theta_{t}),Y_{t})=\tilde{h}_{t}(\tilde{g}_{\theta,t}(\theta_{t}),Y_{t},\alpha)$

which implies that $\tilde{h}_{t}$ is identified. We can now write

$\tilde{\eta}_{I,t}=\tilde{h}^{-1}_{t}(\tilde{g}_{I,t}(I_{t}),\tilde{g}_{\theta,t}(\theta_{t}),Y_{t})$

and therefore the joint distribution of
$(\tilde{g}_{I,t}(I_{t}),\tilde{g}_{\theta,t}(\theta_{t}),Y_{t},\tilde{\eta}_{I,t})$
is point identified.

Now write

$\theta_{t+1}=f_{t}(\theta_{t},I_{t},\eta_{I,t},\varsigma_{\theta,t})\quad\Leftrightarrow\quad\tilde{g}_{\theta,t+1}(\theta_{t+1})=\tilde{g}_{\theta,t+1}(f_{t}(\tilde{g}_{\theta,t}^{-1}(\tilde{g}_{\theta,t}(\theta_{t})),\tilde{g}_{I,t}^{-1}(\tilde{g}_{I,t}(I_{t})),F^{-1}_{\eta_{I,t}}(\tilde{\eta}_{I,t}),\varsigma_{\theta,t}).$

We can therefore identify

$\displaystyle
Q_{{\alpha_{4}}}(\tilde{g}_{\theta,t+1}(\theta_{t+1})\mid\tilde{g}_{\theta,t}(\theta_{t})=Q_{\alpha_{1}}(\tilde{g}_{\theta,t}(\theta_{t})),\tilde{g}_{I,t}(I_{t})=Q_{\alpha_{2}}(\tilde{g}_{I,t}(I_{t})),\tilde{\eta}_{I,t}=\alpha_{3})$
$\displaystyle=\tilde{g}_{\theta,t+1}(f_{t}(\tilde{g}_{\theta,t}^{-1}(Q_{\alpha_{1}}(\tilde{g}_{\theta,t}(\theta_{t})),\tilde{g}_{I,t}^{-1}(Q_{\alpha_{2}}(\tilde{g}_{I,t}(I_{t})),F^{-1}_{\eta_{I,t}}(\alpha_{3}),Q_{\alpha_{4}}(\varsigma_{\theta,t})))$
$\displaystyle=\tilde{g}_{\theta,t+1}(f_{t}(Q_{\alpha_{1}}(\theta_{t}),Q_{\alpha_{2}}(I_{t}),Q_{\alpha_{3}}(\eta_{I,t}),Q_{\alpha_{4}}(\varsigma_{\theta,t})))$

and

$F_{\tilde{g}_{\theta,t+1}(\theta_{t+1})}(\tilde{g}_{\theta,t+1}(f_{t}(Q_{\alpha_{1}}(\theta_{t}),Q_{\alpha_{2}}(I_{t}),Q_{\alpha_{3}}(\eta_{I,t}),Q_{\alpha_{4}}(\varsigma_{\theta,t}))))=F_{\theta_{t+1}}(f_{t}(Q_{\alpha_{1}}(\theta_{t}),Q_{\alpha_{2}}(I_{t}),Q_{\alpha_{3}}(\eta_{I,t}),Q_{\alpha_{4}}(\varsigma_{\theta,t})))$

For the second part notice that

$\displaystyle\tilde{g}_{I,t}(I_{t}(Y))$
$\displaystyle=\tilde{g}_{I,t}(h_{t}(\tilde{g}_{\theta,t+1}^{-1}(Q_{\alpha_{1}}(\tilde{g}_{\theta,t}(\theta_{t}))),Y_{t},F^{-1}_{\eta_{I,t}}(\alpha_{3})))$
$\displaystyle=\tilde{h}_{t}(Q_{\alpha_{1}}(\tilde{g}_{\theta,t}(\theta_{t})),Y_{t},\alpha_{3})$

is identified by the previous arguments. We can therefore identify

$\displaystyle
Q_{\alpha_{4}}(\tilde{g}_{\theta,t+1}(\theta_{t+1})\mid\tilde{g}_{\theta,t}(\theta_{t})=Q_{\alpha_{1}}(\tilde{g}_{\theta,t}(\theta_{t})),\tilde{g}_{I,t}(I_{t})=\tilde{g}_{I,t}(I_{t}(Y)),\tilde{\eta}_{I,t}=\alpha_{3})$
$\displaystyle=\tilde{g}_{\theta,t+1}(f_{t}(\tilde{g}_{\theta,t}^{-1}(Q_{\alpha_{1}}(\tilde{g}_{\theta,t}(\theta_{t})),\tilde{g}_{I,t}^{-1}(\tilde{g}_{I,t}(I_{t}(Y))),F^{-1}_{\eta_{I,t}}(\alpha_{3}),Q_{\alpha_{4}}(\varsigma_{\theta,t})))$
$\displaystyle=\tilde{g}_{\theta,t+1}(f_{t}(Q_{\alpha_{1}}(\theta_{t}),I_{t}(Y),Q_{\alpha_{2}}(\eta_{I,t}),Q_{\alpha_{4}}(\varsigma_{\theta,t})))$

and

$\displaystyle
F_{\tilde{g}_{\theta,t+1}(\theta_{t+1})}(\tilde{g}_{\theta,t+1}(f_{t}(Q_{\alpha_{1}}(\theta_{t}),I_{t}(Y),Q_{\alpha_{2}}(\eta_{I,t}),Q_{\alpha_{4}}(\varsigma_{\theta,t}))))$
$\displaystyle=F_{\theta_{t+1}}(f_{t}(Q_{\alpha_{1}}(\theta_{t}),I_{t}(Y),Q_{\alpha_{2}}(\eta_{I,t}),Q_{\alpha_{4}}(\varsigma_{\theta,t})))$

For the third part notice that

$\tilde{g}_{\theta,t+1}(\theta_{t+1})=\tilde{g}_{\theta,t+1}(f_{t}(\tilde{g}_{\theta,t}^{-1}(\tilde{g}_{\theta,t}(\theta_{t})),\tilde{g}_{I,t}^{-1}(\tilde{g}_{I,t}(I_{t})),F^{-1}_{\eta_{I,t}}(\tilde{\eta}_{I,t}),F_{\varsigma_{\theta,t}}^{-1}(\tilde{\varsigma}_{\theta,t}))).$

with
$\tilde{\varsigma}_{\theta,t}=F_{\varsigma_{\theta,t}}({\varsigma}_{\theta,t})\sim
U[0,1]$. We can therefore write $\tilde{\varsigma}_{\theta,t}$ as an
identified function of the random vector
$(\tilde{g}_{\theta,t+1}(\theta_{t+1}),\tilde{g}_{\theta,t}(\theta_{t}),\tilde{g}_{I,t}(I_{t}),\tilde{\eta}_{I,t})$.
Hence, the joint distribution of

$(Q,\tilde{g}_{\theta,t}(\theta_{t}),\\{\tilde{g}_{I,t}(I_{t})\\}^{T-1}_{s=t},\\{\tilde{\eta}_{I,s}\\}^{T-1}_{s=t},\\{\tilde{\varsigma}_{\theta,s}\\}^{T-1}_{s=t})$

is identified. Finally,

$\displaystyle P\left(Q\leq
q\mid\theta_{t}=Q_{\alpha_{1}}(\theta_{t}),\\{I_{s}=Q_{\alpha_{2s}}(I_{s})\\}^{T-1}_{s=t},\\{\eta_{I,s}=Q_{\alpha_{3s}}(\eta_{I,s})\\}^{T-1}_{s=t},\\{\varsigma_{\theta,s}=Q_{\alpha_{4s}}(\varsigma_{\theta,s})\\}^{T-1}_{s=t}\right)$
$\displaystyle=P\left(Q\leq
q\mid\tilde{\theta}_{t}=Q_{\alpha_{1}}(\tilde{\theta}_{t}),\\{\tilde{g}_{I,t}(I_{t})=Q_{\alpha_{2s}}(\tilde{g}_{I,t}(I_{t}))\\}^{T-1}_{s=t},\\{\tilde{\eta}_{I,s}=Q_{\alpha_{3s}}(\tilde{\eta}_{I,s})\\}^{T-1}_{s=t},\\{\tilde{\varsigma}_{\theta,s}=Q_{\alpha_{4s}}(\tilde{\varsigma}_{\theta,s})\\}^{T-1}_{s=t}\right).$

The fourth part follows from identification of the joint distribution of
$(Q,\tilde{g}_{\theta,t}(\theta_{t}),\\{Y_{s}\\}^{T-1}_{s=t})$.

∎

###### Proof of Theorem 6.

Let $\bar{\mu}_{\theta,0,1}=0$, $\bar{\lambda}_{\theta,0,1}=1$ and
$\ln\bar{\theta}_{0}=\mu_{\theta,0,1}+\lambda_{\theta,0,1}\ln\theta_{0}$. Let
$\bar{\mu}_{\theta,t,1}$ and $\bar{\lambda}_{\theta,t,1}$ and
$\ln\bar{\theta}_{t}$ be such that

$\bar{\mu}_{\theta,t,1}+\bar{\lambda}_{\theta,t,1}\ln\bar{\theta}_{t}=\mu_{\theta,t,1}+\lambda_{\theta,t,1}\ln\theta_{t}.$

These values are not unique for $t>0$ and will be determined by the other
assumptions. Now let
$\bar{\lambda}_{\theta,t,m}=\bar{\lambda}_{\theta,t,1}\frac{{\lambda}_{\theta,t,m}}{{\lambda}_{\theta,t,1}}$
and
$\bar{\mu}_{\theta,t,m}=\mu_{\theta,t,m}-\frac{\lambda_{\theta,t,m}}{\lambda_{\theta,t,1}}\mu_{\theta,t,1}+\frac{\bar{\lambda}_{\theta,t,m}}{\bar{\lambda}_{\theta,t,1}}\bar{\mu}_{\theta,t,1}$
in which case

$\bar{\mu}_{\theta,t,m}+\bar{\lambda}_{\theta,t,m}\ln\bar{\theta}_{t}=\mu_{\theta,t,m}+\lambda_{\theta,t,m}\ln\theta_{t}.$

for all $m$.

Next suppose that $\bar{\mu}_{I,t,1}=0$ and $\bar{\lambda}_{I,t,1}=1$ for all
$t$ and define

$\ln\bar{I}_{t}=\frac{1}{\bar{\lambda}_{I,t,1}}\left(\mu_{I,t,1}-\bar{\mu}_{I,t,1}+\lambda_{I,t,1}\ln
I_{t}\right).$

and
$\bar{\lambda}_{I,t,m}=\bar{\lambda}_{I,t,1}\frac{{\lambda}_{I,t,m}}{{\lambda}_{I,t,1}}$
and
$\bar{\mu}_{I,t,m}=\mu_{I,t,m}-\frac{\lambda_{I,t,m}}{\lambda_{I,t,1}}\mu_{I,t,1}+\frac{\bar{\lambda}_{I,t,m}}{\bar{\lambda}_{I,t,1}}\bar{\mu}_{I,t,1}$
in which case

$\bar{\mu}_{I,t,m}+\bar{\lambda}_{I,t,m}\ln\bar{I}_{t}=\mu_{I,t,m}+\lambda_{I,t,m}\ln
I_{t}$

for all $m$. Moreover, it is easy to see that there are
$(\bar{\beta}_{0t},\bar{\beta}_{1t},\bar{\beta}_{2t})$ and $\bar{\eta}_{I,t}$
such that

$\ln\bar{I}_{t}=\bar{\beta}_{0t}+\bar{\beta}_{1t}\ln\theta_{t}+\bar{\beta}_{2t}\ln
Y_{t}+\bar{\eta}_{I,t}$

Instead of assuming $\bar{\mu}_{I,t,1}=0$ and $\bar{\lambda}_{I,t,1}=1$, first
write

$\displaystyle\ln I_{t}$
$\displaystyle=\beta_{0t}+\beta_{1t}\left(\frac{\bar{\mu}_{\theta,t,1}-\mu_{\theta,t,1}}{\lambda_{\theta,t,1}}+\frac{\bar{\lambda}_{\theta,t,1}}{\lambda_{\theta,t,1}}\ln\bar{\theta}_{t}\right)+\beta_{2t}\ln
Y_{t}+\eta_{I,t}$
$\displaystyle=\beta_{0t}+\beta_{1t}\left(\frac{\bar{\mu}_{\theta,t,1}-\mu_{\theta,t,1}}{\lambda_{\theta,t,1}}\right)+\beta_{1t}\frac{\bar{\lambda}_{\theta,t,1}}{\lambda_{\theta,t,1}}\ln\bar{\theta}_{t}+\beta_{2t}\ln
Y_{t}+\eta_{I,t}$

which implies that

$\displaystyle\ln\bar{I}_{t}$
$\displaystyle=\bar{\beta}_{1}\ln\bar{\theta}_{t}+\bar{\beta}_{2}\ln
Y_{t}+\bar{\eta}_{I,t}$

with

$\ln\bar{I}_{t}=\frac{\ln
I_{t}-\left(\beta_{0t}+\beta_{1t}\left(\frac{\bar{\mu}_{\theta,t,1}-\mu_{\theta,t,1}}{\lambda_{\theta,t,1}}\right)\right)}{\beta_{1t}\frac{\bar{\lambda}_{\theta,t,1}}{\lambda_{\theta,t,1}}+\beta_{2t}},$
$\bar{\beta}_{1t}=\frac{\beta_{1t}\frac{\bar{\lambda}_{\theta,t,1}}{\lambda_{\theta,t,1}}}{\beta_{1t}\frac{\bar{\lambda}_{\theta,t,1}}{\lambda_{\theta,t,1}}+\beta_{2t}}\qquad\text{
and
}\qquad\bar{\beta}_{2t}=\frac{\beta_{2t}}{\beta_{1t}\frac{\bar{\lambda}_{\theta,t,1}}{\lambda_{\theta,t,1}}+\beta_{2t}}$

It is now easy to see that there exist $\bar{\mu}_{I,t,1}$ and
$\bar{\lambda}_{I,t,1}$ such that

$\bar{\mu}_{I,t,1}+\bar{\lambda}_{I,t,1}\ln\bar{I}_{t}=\mu_{I,t,1}+\lambda_{I,t,1}\ln
I_{t}.$

Then define and
$\bar{\lambda}_{I,t,m}=\bar{\lambda}_{I,t,1}\frac{{\lambda}_{I,t,m}}{{\lambda}_{I,t,1}}$
and
$\bar{\mu}_{I,t,m}=\mu_{I,t,m}-\frac{\lambda_{I,t,m}}{\lambda_{I,t,1}}\mu_{I,t,1}+\frac{\bar{\lambda}_{I,t,m}}{\bar{\lambda}_{I,t,1}}\bar{\mu}_{I,t,1}$
in which case

$\bar{\mu}_{I,t,m}+\bar{\lambda}_{I,t,m}\ln\bar{I}_{t}=\mu_{I,t,m}+\lambda_{I,t,m}\ln
I_{t}$

for all $m$.

In both cases, there are parameters that consistent with the second, third,
and forth equation of the model. In addition, we now have known constant
$\bar{\mu}_{I,t,1}$ and $\bar{\lambda}_{I,t,1}$ and a random variable
$\ln\bar{I}_{t}$ such that

$\bar{\mu}_{I,t,1}+\bar{\lambda}_{I,t,1}\ln\bar{\theta}_{t}=\mu_{I,t,1}+\lambda_{I,t,1}\ln
I_{t}.$

We next show that there production function parameters that are consistent
with Assumption 3(b). To do so, write

$\displaystyle\ln\theta_{t+1}$
$\displaystyle=a_{t}+\gamma_{1t}\ln\theta_{t}+\gamma_{2t}\ln
I_{t}+\gamma_{3t}\ln\theta_{t}\ln I_{t}+\eta_{\theta,t}$
$\displaystyle=a_{t}+\gamma_{1t}\left(\frac{\bar{\mu}_{\theta,t,1}-\mu_{\theta,t,1}}{\lambda_{\theta,t,1}}+\frac{\bar{\lambda}_{\theta,t,1}}{\lambda_{\theta,t,1}}\ln\bar{\theta}_{t}\right)+\gamma_{2t}\left(\frac{\bar{\mu}_{I,t,1}-\mu_{I,t,1}}{\lambda_{I,t,1}}+\frac{\bar{\lambda}_{I,t,1}}{\lambda_{I,t,1}}\ln\bar{I}_{t}\right)$
$\displaystyle\qquad+\gamma_{3t}\left(\frac{\bar{\mu}_{\theta,t,1}-\mu_{\theta,t,1}}{\lambda_{\theta,t,1}}+\frac{\bar{\lambda}_{\theta,t,1}}{\lambda_{\theta,t,1}}\ln\bar{\theta}_{t}\right)\left(\frac{\bar{\mu}_{I,t,1}-\mu_{I,t,1}}{\lambda_{I,t,1}}+\frac{\bar{\lambda}_{I,t,1}}{\lambda_{I,t,1}}\ln\bar{I}_{t}\right)+\eta_{\theta,t}$
$\displaystyle=a_{t}+\gamma_{1t}\frac{\bar{\mu}_{\theta,t,1}-\mu_{\theta,t,1}}{\lambda_{\theta,t,1}}+\gamma_{2t}\frac{\bar{\mu}_{I,t,1}-\mu_{I,t,1}}{\lambda_{I,t,1}}+\gamma_{3t}\frac{\bar{\mu}_{\theta,t,1}-\mu_{\theta,t,1}}{\lambda_{\theta,t,1}}\frac{\bar{\mu}_{I,t,1}-\mu_{I,t,1}}{\lambda_{I,t,1}}$
$\displaystyle\qquad+\left(\gamma_{1t}\frac{\bar{\lambda}_{\theta,t,1}}{\lambda_{\theta,t,1}}+\gamma_{3t}\frac{\bar{\mu}_{I,t,1}-\mu_{I,t,1}}{\lambda_{I,t,1}}\frac{\bar{\lambda}_{\theta,t,1}}{\lambda_{\theta,t,1}}\right)\ln\bar{\theta}_{t}+\left(\gamma_{2t}\frac{\bar{\lambda}_{I,t,1}}{\lambda_{I,t,1}}+\gamma_{3t}\frac{\bar{\mu}_{\theta,t,1}-\mu_{\theta,t,1}}{\lambda_{\theta,t,1}}\frac{\bar{\lambda}_{I,t,1}}{\lambda_{I,t,1}}\right)\ln\bar{I}_{t}$
$\displaystyle\qquad+\gamma_{3t}\frac{\bar{\lambda}_{\theta,t,1}}{\lambda_{\theta,t,1}}\frac{\bar{\lambda}_{I,t,1}}{\lambda_{I,t,1}}\ln\bar{\theta}_{t}\ln\bar{I}_{t}+\eta_{\theta,t}$

which we can write as

$\displaystyle\ln\bar{\theta}_{t+1}$
$\displaystyle=\bar{\gamma}_{1t}\ln\bar{\theta}_{t}+\bar{\gamma}_{2t}\ln\bar{I}_{t}+\bar{\gamma}_{3t}\ln\bar{\theta}_{t}\ln\bar{I}_{t}+\bar{\eta}_{\theta,t}$

where

$\ln\bar{\theta}_{t+1}=\frac{\ln\theta_{t+1}-\left(a_{t}+\gamma_{1t}\frac{\bar{\mu}_{\theta,t,1}-\mu_{\theta,t,1}}{\lambda_{\theta,t,1}}+\gamma_{2t}\frac{\bar{\mu}_{I,t,1}-\mu_{I,t,1}}{\lambda_{I,t,1}}+\gamma_{3t}\frac{\bar{\mu}_{\theta,t,1}-\mu_{\theta,t,1}}{\lambda_{\theta,t,1}}\frac{\bar{\mu}_{I,t,1}-\mu_{I,t,1}}{\lambda_{I,t,1}}\right)}{\gamma_{1t}\frac{\bar{\lambda}_{\theta,t,1}}{\lambda_{\theta,t,1}}+\gamma_{3t}\frac{\bar{\mu}_{I,t,1}-\mu_{I,t,1}}{\lambda_{I,t,1}}\frac{\bar{\lambda}_{\theta,t,1}}{\lambda_{\theta,t,1}}+\gamma_{2t}\frac{\bar{\lambda}_{I,t,1}}{\lambda_{I,t,1}}+\gamma_{3t}\frac{\bar{\mu}_{\theta,t,1}-\mu_{\theta,t,1}}{\lambda_{\theta,t,1}}\frac{\bar{\lambda}_{I,t,1}}{\lambda_{I,t,1}}++\gamma_{3t}\frac{\bar{\lambda}_{\theta,t,1}}{\lambda_{\theta,t,1}}\frac{\bar{\lambda}_{I,t,1}}{\lambda_{I,t,1}}}$
$\bar{\gamma}_{1t}=\frac{\gamma_{1t}\frac{\bar{\lambda}_{\theta,t,1}}{\lambda_{\theta,t,1}}+\gamma_{3t}\frac{\bar{\mu}_{I,t,1}-\mu_{I,t,1}}{\lambda_{I,t,1}}\frac{\bar{\lambda}_{\theta,t,1}}{\lambda_{\theta,t,1}}}{\gamma_{1t}\frac{\bar{\lambda}_{\theta,t,1}}{\lambda_{\theta,t,1}}+\gamma_{3t}\frac{\bar{\mu}_{I,t,1}-\mu_{I,t,1}}{\lambda_{I,t,1}}\frac{\bar{\lambda}_{\theta,t,1}}{\lambda_{\theta,t,1}}+\gamma_{2t}\frac{\bar{\lambda}_{I,t,1}}{\lambda_{I,t,1}}+\gamma_{3t}\frac{\bar{\mu}_{\theta,t,1}-\mu_{\theta,t,1}}{\lambda_{\theta,t,1}}\frac{\bar{\lambda}_{I,t,1}}{\lambda_{I,t,1}}++\gamma_{3t}\frac{\bar{\lambda}_{\theta,t,1}}{\lambda_{\theta,t,1}}\frac{\bar{\lambda}_{I,t,1}}{\lambda_{I,t,1}}}$
$\bar{\gamma}_{2t}=\frac{\gamma_{2t}\frac{\bar{\lambda}_{I,t,1}}{\lambda_{I,t,1}}+\gamma_{3t}\frac{\bar{\mu}_{\theta,t,1}-\mu_{\theta,t,1}}{\lambda_{\theta,t,1}}\frac{\bar{\lambda}_{I,t,1}}{\lambda_{I,t,1}}}{\gamma_{1t}\frac{\bar{\lambda}_{\theta,t,1}}{\lambda_{\theta,t,1}}+\gamma_{3t}\frac{\bar{\mu}_{I,t,1}-\mu_{I,t,1}}{\lambda_{I,t,1}}\frac{\bar{\lambda}_{\theta,t,1}}{\lambda_{\theta,t,1}}+\gamma_{2t}\frac{\bar{\lambda}_{I,t,1}}{\lambda_{I,t,1}}+\gamma_{3t}\frac{\bar{\mu}_{\theta,t,1}-\mu_{\theta,t,1}}{\lambda_{\theta,t,1}}\frac{\bar{\lambda}_{I,t,1}}{\lambda_{I,t,1}}++\gamma_{3t}\frac{\bar{\lambda}_{\theta,t,1}}{\lambda_{\theta,t,1}}\frac{\bar{\lambda}_{I,t,1}}{\lambda_{I,t,1}}}$
$\bar{\gamma}_{3t}=\frac{\gamma_{3t}\frac{\bar{\lambda}_{\theta,t,1}}{\lambda_{\theta,t,1}}\frac{\bar{\lambda}_{I,t,1}}{\lambda_{I,t,1}}}{\gamma_{1t}\frac{\bar{\lambda}_{\theta,t,1}}{\lambda_{\theta,t,1}}+\gamma_{3t}\frac{\bar{\mu}_{I,t,1}-\mu_{I,t,1}}{\lambda_{I,t,1}}\frac{\bar{\lambda}_{\theta,t,1}}{\lambda_{\theta,t,1}}+\gamma_{2t}\frac{\bar{\lambda}_{I,t,1}}{\lambda_{I,t,1}}+\gamma_{3t}\frac{\bar{\mu}_{\theta,t,1}-\mu_{\theta,t,1}}{\lambda_{\theta,t,1}}\frac{\bar{\lambda}_{I,t,1}}{\lambda_{I,t,1}}++\gamma_{3t}\frac{\bar{\lambda}_{\theta,t,1}}{\lambda_{\theta,t,1}}\frac{\bar{\lambda}_{I,t,1}}{\lambda_{I,t,1}}}$

It is now easy to see that there exist $\bar{\mu}_{\theta,t+1,1}$ and
$\bar{\lambda}_{\theta,t+1,1}$ such that

$\bar{\mu}_{\theta,t+1,1}+\bar{\lambda}_{\theta,t+1,1}\ln\bar{\theta}_{t+1}=\mu_{\theta,t+1,1}+\lambda_{\theta,t+1,1}\ln\theta_{t+1}.$

We can now use the arguments recursively and show always exist sets of
parameters for the first four equations that are consistent with the data and
satisfy Assumptions 1, 2, 3(b), and either 4(a) or 4(b). It also immediately
follows that the exist $\bar{\rho}_{0}$ and $\bar{\rho}_{1}$ such that

$\bar{\rho}_{0}+\bar{\rho}_{1}\ln\theta_{T}=\rho_{0}+\rho_{1}\ln\theta_{T}$

The arguments in Section 3.2.1 imply that there always exist sets of
parameters that are consistent with the data and satisfy that Assumptions 1,
2, either 3(a) and 4(a).

Finally, suppose Assumptions 1, 2, either 3(a) and 4(b) hold. Let

$\ln\bar{\theta}_{t}=\mu_{\theta,t,1}+\lambda_{\theta,t,1}\ln\theta_{t}.$

Then we can write

$\displaystyle\ln\bar{I}_{t}$
$\displaystyle=\bar{\beta}_{1}\ln\bar{\theta}_{t}+\bar{\beta}_{2}\ln
Y_{t}+\bar{\eta}_{I,t}$

with

$\ln\bar{I}_{t}=\frac{\ln
I_{t}-\left(\beta_{0t}+\beta_{1t}\left(\frac{\mu_{\theta,t,1}}{\lambda_{\theta,t,1}}\right)\right)}{\beta_{1t}\frac{1}{\lambda_{\theta,t,1}}+\beta_{2t}},$
$\bar{\beta}_{1t}=\frac{\beta_{1t}\frac{1}{\lambda_{\theta,t,1}}}{\beta_{1t}\frac{1}{\lambda_{\theta,t,1}}+\beta_{2t}}\qquad\text{
and
}\qquad\bar{\beta}_{2t}=\frac{\beta_{2t}}{\beta_{1t}\frac{1}{\lambda_{\theta,t,1}}+\beta_{2t}}$

It is now easy to see that there exist $\bar{\mu}_{I,t,1}$ and
$\bar{\lambda}_{I,t,1}$ such that

$\bar{\mu}_{I,t,1}+\bar{\lambda}_{I,t,1}\ln\bar{I}_{t}=\mu_{I,t,1}+\lambda_{I,t,1}\ln
I_{t}.$

Using the previous arguments, we can then write

$\displaystyle\ln\bar{\theta}_{t+1}$
$\displaystyle=\mu_{\theta,t+1,1}+\lambda_{\theta,t+1,1}\left(a_{t}+\gamma_{1t}\frac{\bar{\mu}_{\theta,t,1}-\mu_{\theta,t,1}}{\lambda_{\theta,t,1}}+\gamma_{2t}\frac{\bar{\mu}_{I,t,1}-\mu_{I,t,1}}{\lambda_{I,t,1}}+\gamma_{3t}\frac{\bar{\mu}_{\theta,t,1}-\mu_{\theta,t,1}}{\lambda_{\theta,t,1}}\frac{\bar{\mu}_{I,t,1}-\mu_{I,t,1}}{\lambda_{I,t,1}}\right)$
$\displaystyle\quad+\lambda_{\theta,t+1,1}\left(\gamma_{1t}\frac{\bar{\lambda}_{\theta,t,1}}{\lambda_{\theta,t,1}}+\gamma_{3t}\frac{\bar{\mu}_{I,t,1}-\mu_{I,t,1}}{\lambda_{I,t,1}}\frac{\bar{\lambda}_{\theta,t,1}}{\lambda_{\theta,t,1}}\right)\ln\bar{\theta}_{t}+\lambda_{\theta,t+1,1}\left(\gamma_{2t}\frac{\bar{\lambda}_{I,t,1}}{\lambda_{I,t,1}}+\gamma_{3t}\frac{\bar{\mu}_{\theta,t,1}-\mu_{\theta,t,1}}{\lambda_{\theta,t,1}}\frac{\bar{\lambda}_{I,t,1}}{\lambda_{I,t,1}}\right)\ln\bar{I}_{t}$
$\displaystyle\quad+\lambda_{\theta,t+1,1}\gamma_{3t}\frac{\bar{\lambda}_{\theta,t,1}}{\lambda_{\theta,t,1}}\frac{\bar{\lambda}_{I,t,1}}{\lambda_{I,t,1}}\ln\bar{\theta}_{t}\ln\bar{I}_{t}+\lambda_{\theta,t+1,1}\eta_{\theta,t}$

Setting $\bar{\mu}_{\theta,t,1}=0$ and $\bar{\lambda}_{\theta,t,1}=1$, it is
easy to that there are $\bar{a}_{t}$, $\bar{\gamma}_{1t}$,
$\bar{\gamma}_{2t}$, and $\bar{\gamma}_{3t}$ such that such that

$\ln\bar{\theta}_{t+1}=\bar{a}_{t}+\bar{\gamma}_{1t}\ln\bar{\theta}_{t}+\bar{\gamma}_{2t}\ln\bar{I}_{t}+\bar{\gamma}_{3t}\ln\bar{\theta}_{t}\ln\bar{I}_{t}+\bar{\eta}_{\theta,t}$

∎

###### Proof of Theorem 7.

As in the proof of Theorem 6, there are constants $\bar{\mu}_{\theta,t,1}$ and
$\bar{\lambda}_{\theta,t,1}$ and a random variable $\ln\bar{\theta}_{t}$ such
that

$\bar{\mu}_{\theta,t,1}+\bar{\lambda}_{\theta,t,1}\ln\bar{\theta}_{t}=\mu_{\theta,t,1}+\lambda_{\theta,t,1}\ln\theta_{t}$

and $\bar{\mu}_{\theta,0,1}=0$ and $\bar{\lambda}_{\theta,0,1}=1$. Then there
exist $\bar{\mu}_{\theta,t,m}$ and $\bar{\lambda}_{\theta,t,m}$ such that for
all $m$

$\bar{\mu}_{\theta,t,m}+\bar{\lambda}_{\theta,t,m}\ln\bar{\theta}_{t}=\mu_{\theta,t,m}+\lambda_{\theta,t,m}\ln\theta_{t}$

Next consider the production function and write

$\displaystyle\theta_{t+1}$
$\displaystyle=\left(\gamma_{1t}\theta_{t}^{\sigma_{t}}+\gamma_{2t}I_{t}^{\sigma_{t}}\right)^{\frac{\psi_{t}}{\sigma_{t}}}\exp(\eta_{\theta,t})$
$\displaystyle=\left(\gamma_{1t}\exp\left({\sigma_{t}}\frac{\bar{\mu}_{\theta,t,1}-\mu_{\theta,t,1}}{\lambda_{\theta,t,1}}\right)\bar{\theta}_{t}^{\sigma_{t}\frac{\bar{\lambda}_{\theta,t,1}}{\lambda_{\theta,t,1}}}+\gamma_{2t}I_{t}^{\sigma_{t}}\right)^{\frac{\psi_{t}}{\sigma_{t}}}\exp(\eta_{\theta,t})$
$\displaystyle=\left(\gamma_{1t}\exp\left({\sigma_{t}}\frac{\bar{\mu}_{\theta,t,1}-\mu_{\theta,t,1}}{\lambda_{\theta,t,1}}\right)\bar{\theta}_{t}^{\sigma_{t}\frac{\bar{\lambda}_{\theta,t,1}}{\lambda_{\theta,t,1}}}+\gamma_{2t}\left(I_{t}^{\frac{\lambda_{\theta,t,1}}{\bar{\lambda}_{\theta,t,1}}}\right)^{\sigma_{t}\frac{\bar{\lambda}_{\theta,t,1}}{\lambda_{\theta,t,1}}}\right)^{\frac{\psi_{t}}{\sigma_{t}}}\exp(\eta_{\theta,t})$

and with
$\bar{\sigma}_{t}=\sigma_{t}\frac{\bar{\lambda}_{\theta,t,1}}{\lambda_{\theta,t,1}}$
we have

$\displaystyle\theta_{t+1}^{\frac{\lambda_{\theta,t,1}}{\bar{\lambda}_{\theta,t,1}}}$
$\displaystyle=\left(\gamma_{1t}\exp\left({\sigma_{t}}\frac{\bar{\mu}_{\theta,t,1}-\mu_{\theta,t,1}}{\lambda_{\theta,t,1}}\right)\bar{\theta}_{t}^{\bar{\sigma}_{t}}+\gamma_{2t}\left(I_{t}^{\frac{\lambda_{\theta,t,1}}{\bar{\lambda}_{\theta,t,1}}}\right)^{\bar{\sigma}_{t}}\right)^{\frac{\psi_{t}}{\bar{\sigma}_{t}}}\exp\left(\frac{\lambda_{\theta,t,1}}{\bar{\lambda}_{\theta,t,1}}\eta_{\theta,t}\right)$

We also know that we need to satisfy the relationship

$\exp\left(\frac{\bar{\mu}_{I,t,1}-\mu_{I,t,1}}{\lambda_{I,t,1}}+\frac{\bar{\lambda}_{I,t,1}}{\lambda_{I,t,1}}\ln\bar{I}_{t}\right)=I_{t}$

Hence,

$\displaystyle\theta_{t+1}^{\frac{\lambda_{\theta,t,1}}{\bar{\lambda}_{\theta,t,1}}}$
$\displaystyle=\left(\gamma_{1t}\exp\left({\sigma_{t}}\frac{\bar{\mu}_{\theta,t,1}-\mu_{\theta,t,1}}{\lambda_{\theta,t,1}}\right)\bar{\theta}_{t}^{\bar{\sigma}_{t}}+\gamma_{2t}\exp\left(\bar{\sigma}_{t}\frac{\lambda_{\theta,t,1}}{\bar{\lambda}_{\theta,t,1}}\frac{\bar{\mu}_{I,t,1}-\mu_{I,t,1}}{\lambda_{I,t,1}}\right)\bar{I}_{t}^{\bar{\sigma}_{t}\frac{\lambda_{\theta,t,1}}{\bar{\lambda}_{\theta,t,1}}\frac{\bar{\lambda}_{I,t,1}}{\lambda_{I,t,1}}}\right)^{\frac{\psi_{t}}{\bar{\sigma}_{t}}}\exp\left(\frac{\lambda_{\theta,t,1}}{\bar{\lambda}_{\theta,t,1}}\eta_{\theta,t}\right)$

Since

$\theta_{t+1}^{\frac{\lambda_{\theta,t,1}}{\bar{\lambda}_{\theta,t,1}}}=\exp\left(\frac{\lambda_{\theta,t,1}}{\bar{\lambda}_{\theta,t,1}}\frac{\bar{\mu}_{\theta,t+1,1}-\mu_{\theta,t+1,1}}{\lambda_{\theta,t+1,1}}\right)\bar{\theta}_{t+1}^{\frac{\lambda_{\theta,t,1}}{\bar{\lambda}_{\theta,t,1}}\frac{\bar{\lambda}_{\theta,t+1,1}}{\lambda_{\theta,t+1,1}}}$

we then write

$\displaystyle\bar{\theta}_{t+1}$
$\displaystyle=\left(\frac{\gamma_{1t}\exp\left({\sigma_{t}}\frac{\bar{\mu}_{\theta,t,1}-\mu_{\theta,t,1}}{\lambda_{\theta,t,1}}\right)}{\exp\left(\bar{\sigma}_{t}\frac{\lambda_{\theta,t,1}}{\bar{\lambda}_{\theta,t,1}}\frac{\bar{\mu}_{\theta,t+1,1}-\mu_{\theta,t+1,1}}{\lambda_{\theta,t+1,1}}\right)}\bar{\theta}_{t}^{\bar{\sigma}_{t}}+\frac{\gamma_{2t}\exp\left(\bar{\sigma}_{t}\frac{\lambda_{\theta,t,1}}{\bar{\lambda}_{\theta,t,1}}\frac{\bar{\mu}_{I,t,1}-\mu_{I,t,1}}{\lambda_{I,t,1}}\right)}{\exp\left(\bar{\sigma}_{t}\frac{\lambda_{\theta,t,1}}{\bar{\lambda}_{\theta,t,1}}\frac{\bar{\mu}_{\theta,t+1,1}-\mu_{\theta,t+1,1}}{\lambda_{\theta,t+1,1}}\right)}\bar{I}_{t}^{\bar{\sigma}_{t}\frac{\lambda_{\theta,t,1}}{\bar{\lambda}_{\theta,t,1}}\frac{\bar{\lambda}_{I,t,1}}{\lambda_{I,t,1}}}\right)^{\frac{\bar{\psi}_{t}}{\bar{\sigma}_{t}}}\exp\left(\frac{\lambda_{\theta,t,1}}{\bar{\lambda}_{\theta,t,1}}\eta_{\theta,t}\right)$
$\displaystyle=\left(\bar{\gamma}_{1t}\bar{\theta}_{t}^{\bar{\sigma}_{t}}+\bar{\gamma}_{2t}\bar{I}_{t}^{\bar{\sigma}_{t}\frac{\lambda_{\theta,t,1}}{\bar{\lambda}_{\theta,t,1}}\frac{\bar{\lambda}_{I,t,1}}{\lambda_{I,t,1}}}\right)^{\frac{\bar{\psi}_{t}}{\bar{\sigma}_{t}}}\exp\left(\frac{\lambda_{\theta,t,1}}{\bar{\lambda}_{\theta,t,1}}\eta_{\theta,t}\right)$

where

$\bar{\gamma}_{1t}=\frac{\gamma_{1t}\exp\left({\sigma_{t}}\frac{\bar{\mu}_{\theta,t,1}-\mu_{\theta,t,1}}{\lambda_{\theta,t,1}}\right)}{\exp\left(\bar{\sigma}_{t}\frac{\lambda_{\theta,t,1}}{\bar{\lambda}_{\theta,t,1}}\frac{\bar{\mu}_{\theta,t+1,1}-\mu_{\theta,t+1,1}}{\lambda_{\theta,t+1,1}}\right)}\quad\text{
and
}\quad\bar{\gamma}_{2t}=\frac{\gamma_{2t}\exp\left(\bar{\sigma}_{t}\frac{\lambda_{\theta,t,1}}{\bar{\lambda}_{\theta,t,1}}\frac{\bar{\mu}_{I,t,1}-\mu_{I,t,1}}{\lambda_{I,t,1}}\right)}{\exp\left(\bar{\sigma}_{t}\frac{\lambda_{\theta,t,1}}{\bar{\lambda}_{\theta,t,1}}\frac{\bar{\mu}_{\theta,t+1,1}-\mu_{\theta,t+1,1}}{\lambda_{\theta,t+1,1}}\right)}$

and
$\bar{\psi}_{t}=\psi_{t}{\frac{\bar{\lambda}_{\theta,t,1}}{\lambda_{\theta,t,1}}\frac{\lambda_{\theta,t+1,1}}{\bar{\lambda}_{\theta,t+1,1}}}=\psi_{t}{\frac{\bar{\sigma}_{t}}{{\sigma}_{t}}\frac{{\sigma}_{t+1}}{\bar{\sigma}_{t+1}}}$.

We now show that under different combinations of the assumptions, there always
parameters that are consistent with the model. First notice that it has to
hold that

$\bar{\lambda}_{I,t,1}=\frac{\bar{\lambda}_{\theta,t,1}}{\lambda_{\theta,t,1}}\lambda_{I,t,1}$

Now write

$\displaystyle\left(\frac{\bar{\mu}_{I,t,1}-\mu_{I,t,1}}{\lambda_{I,t,1}}+\frac{\bar{\lambda}_{I,t,1}}{\lambda_{I,t,1}}\ln\bar{I}_{t}\right)$
$\displaystyle=\beta_{0t}+\beta_{1t}\left(\frac{\bar{\mu}_{\theta,t,1}-\mu_{\theta,t,1}}{\lambda_{\theta,t,1}}+\frac{\bar{\lambda}_{\theta,t,1}}{\lambda_{\theta,t,1}}\ln\bar{\theta}_{t}\right)+\beta_{2t}\ln
Y_{t}+\eta_{I,t}$

which implies that

$\displaystyle\ln\bar{I}_{t}$
$\displaystyle=\beta_{0t}\frac{\lambda_{I,t,1}}{\bar{\lambda}_{I,t,1}}-\frac{\bar{\mu}_{I,t,1}-\mu_{I,t,1}}{\lambda_{I,t,1}}\frac{\lambda_{I,t,1}}{\bar{\lambda}_{I,t,1}}+\beta_{1t}\frac{\lambda_{I,t,1}}{\bar{\lambda}_{I,t,1}}\frac{\bar{\mu}_{\theta,t,1}-\mu_{\theta,t,1}}{\lambda_{\theta,t,1}}+\beta_{1t}\ln\bar{\theta}_{t}+\beta_{2t}\frac{\lambda_{I,t,1}}{\bar{\lambda}_{I,t,1}}\ln
Y_{t}+\eta_{I,t}$

It follows that $\bar{\beta}_{1}=\beta_{1}$ and
$\bar{\beta}_{2}=\beta_{2t}\frac{\lambda_{I,t,1}}{\bar{\lambda}_{I,t,1}}$ are
determined and

$\bar{\beta}_{0t}=\beta_{0t}\frac{\lambda_{I,t,1}}{\bar{\lambda}_{I,t,1}}-\frac{\bar{\mu}_{I,t,1}-\mu_{I,t,1}}{\lambda_{I,t,1}}\frac{\lambda_{I,t,1}}{\bar{\lambda}_{I,t,1}}+\beta_{1t}\frac{\lambda_{I,t,1}}{\bar{\lambda}_{I,t,1}}\frac{\bar{\mu}_{\theta,t,1}-\mu_{\theta,t,1}}{\lambda_{\theta,t,1}}$

If $\bar{\lambda}_{\theta,t,1}=\bar{\lambda}_{\theta,t+1,1}=1$, set
$\bar{\psi}_{t}=\psi_{t}\frac{\lambda_{\theta,t+1,1}}{\lambda_{\theta,t,1}}$
and $\bar{\sigma}_{t}=\sigma_{t}\frac{1}{\lambda_{\theta,t,1}}$. If instead
$\bar{\psi}_{t}=1$, take
$\bar{\lambda}_{\theta,t+1,1}=\psi_{t}{\frac{\bar{\lambda}_{\theta,t,1}}{\lambda_{\theta,t,1}}{\lambda_{\theta,t+1,1}}}$
and
$\bar{\sigma}_{t}=\sigma_{t}\frac{\bar{\lambda}_{\theta,t,1}}{\lambda_{\theta,t,1}}$.
Either way, $\bar{\sigma}_{t}$, $\bar{\psi}_{t}$, and
$\bar{\lambda}_{\theta,t,1}$ are uniquely determined.

Now suppose that $\bar{\mu}_{\theta,t,1}=\bar{\mu}_{I,t,1}=0$. Then
$\bar{\beta}_{0t}$ is uniquely determined and so is

$\bar{\gamma}_{1t}=\frac{\gamma_{1t}\exp\left({\sigma_{t}}\frac{\bar{\mu}_{\theta,t,1}-\mu_{\theta,t,1}}{\lambda_{\theta,t,1}}\right)}{\exp\left(\bar{\sigma}_{t}\frac{\lambda_{\theta,t,1}}{\bar{\lambda}_{\theta,t,1}}\frac{\bar{\mu}_{\theta,t+1,1}-\mu_{\theta,t+1,1}}{\lambda_{\theta,t+1,1}}\right)}\quad\text{and}\quad\bar{\gamma}_{2t}=\frac{\gamma_{2t}\exp\left(\bar{\sigma}_{t}\frac{\lambda_{\theta,t,1}}{\bar{\lambda}_{\theta,t,1}}\frac{\bar{\mu}_{I,t,1}-\mu_{I,t,1}}{\lambda_{I,t,1}}\right)}{\exp\left(\bar{\sigma}_{t}\frac{\lambda_{\theta,t,1}}{\bar{\lambda}_{\theta,t,1}}\frac{\bar{\mu}_{\theta,t+1,1}-\mu_{\theta,t+1,1}}{\lambda_{\theta,t+1,1}}\right)}.$

Hence, we found parameters that are consistent with the first four equations
of the model.

Next suppose that $\bar{\mu}_{\theta,t,1}=0$ and $\bar{\beta}_{0t}=0$. Then
also $\bar{\mu}_{\theta,t+1,1}=0$. Again let

$\bar{\gamma}_{1t}=\frac{\gamma_{1t}\exp\left({\sigma_{t}}\frac{\bar{\mu}_{\theta,t,1}-\mu_{\theta,t,1}}{\lambda_{\theta,t,1}}\right)}{\exp\left(\bar{\sigma}_{t}\frac{\lambda_{\theta,t,1}}{\bar{\lambda}_{\theta,t,1}}\frac{\bar{\mu}_{\theta,t+1,1}-\mu_{\theta,t+1,1}}{\lambda_{\theta,t+1,1}}\right)}$

Then we can find $\bar{\mu}_{I,t,1}$ such that $\bar{\beta}_{0t}=0$. Finally
let

$\bar{\gamma}_{2t}=\frac{\gamma_{2t}\exp\left(\bar{\sigma}_{t}\frac{\lambda_{\theta,t,1}}{\bar{\lambda}_{\theta,t,1}}\frac{\bar{\mu}_{I,t,1}-\mu_{I,t,1}}{\lambda_{I,t,1}}\right)}{\exp\left(\bar{\sigma}_{t}\frac{\lambda_{\theta,t,1}}{\bar{\lambda}_{\theta,t,1}}\frac{\bar{\mu}_{\theta,t+1,1}-\mu_{\theta,t+1,1}}{\lambda_{\theta,t+1,1}}\right)}.$

Hence, we found parameters that are consistent with the first four equations
of the model.

Next suppose that $\bar{\mu}_{I,t,1}=0$ and $\gamma_{1t}+\gamma_{2t}=1$. Then
$\bar{\beta}_{0t}=\beta_{0t}\frac{\lambda_{I,t,1}}{\bar{\lambda}_{I,t,1}}$ is
uniquely determined. It then has to hold that

$\bar{\gamma}_{1t}=\frac{\gamma_{1t}\exp\left({\sigma_{t}}\frac{\bar{\mu}_{\theta,t,1}-\mu_{\theta,t,1}}{\lambda_{\theta,t,1}}\right)}{\exp\left(\bar{\sigma}_{t}\frac{\lambda_{\theta,t,1}}{\bar{\lambda}_{\theta,t,1}}\frac{\bar{\mu}_{\theta,t+1,1}-\mu_{\theta,t+1,1}}{\lambda_{\theta,t+1,1}}\right)}\quad\text{and}\quad\bar{\gamma}_{2t}=\frac{\gamma_{2t}\exp\left(\bar{\sigma}_{t}\frac{\lambda_{\theta,t,1}}{\bar{\lambda}_{\theta,t,1}}\frac{\bar{\mu}_{I,t,1}-\mu_{I,t,1}}{\lambda_{I,t,1}}\right)}{\exp\left(\bar{\sigma}_{t}\frac{\lambda_{\theta,t,1}}{\bar{\lambda}_{\theta,t,1}}\frac{\bar{\mu}_{\theta,t+1,1}-\mu_{\theta,t+1,1}}{\lambda_{\theta,t+1,1}}\right)}.$

and $\bar{\gamma}_{1t}+\bar{\gamma}_{2t}=1$. Since the denominator of the
fractions is strictly monotone in $\bar{\mu}_{\theta,t+1,1}$ and has range
$(0,\infty)$, there exists a unique value of $\bar{\mu}_{\theta,t+1,1}$ such
that $\bar{\gamma}_{1t}+\bar{\gamma}_{2t}=1$. We then found parameters that
are consistent with the first four equations of the model.

Finally, suppose that $\bar{\beta}_{0t}=0$ and $\gamma_{1t}+\gamma_{2t}=1$.
Then we can find a unique value $\bar{\mu}_{I,t,1}=0$ such that
$\bar{\beta}_{0t}=0$. Given this value, it has to hold that

$\bar{\gamma}_{1t}=\frac{\gamma_{1t}\exp\left({\sigma_{t}}\frac{\bar{\mu}_{\theta,t,1}-\mu_{\theta,t,1}}{\lambda_{\theta,t,1}}\right)}{\exp\left(\bar{\sigma}_{t}\frac{\lambda_{\theta,t,1}}{\bar{\lambda}_{\theta,t,1}}\frac{\bar{\mu}_{\theta,t+1,1}-\mu_{\theta,t+1,1}}{\lambda_{\theta,t+1,1}}\right)}\quad\text{and}\quad\bar{\gamma}_{2t}=\frac{\gamma_{2t}\exp\left(\bar{\sigma}_{t}\frac{\lambda_{\theta,t,1}}{\bar{\lambda}_{\theta,t,1}}\frac{\bar{\mu}_{I,t,1}-\mu_{I,t,1}}{\lambda_{I,t,1}}\right)}{\exp\left(\bar{\sigma}_{t}\frac{\lambda_{\theta,t,1}}{\bar{\lambda}_{\theta,t,1}}\frac{\bar{\mu}_{\theta,t+1,1}-\mu_{\theta,t+1,1}}{\lambda_{\theta,t+1,1}}\right)}.$

and $\bar{\gamma}_{1t}+\bar{\gamma}_{2t}=1$. Again, we there exists a unique
value of $\bar{\mu}_{\theta,t+1,1}$ such that
$\bar{\gamma}_{1t}+\bar{\gamma}_{2t}=1$. We then found parameters that are
consistent with the first four equations of the model.

In all four cases, it also immediately follows that the exist $\bar{\rho}_{0}$
and $\bar{\rho}_{1}$ such that

$\bar{\rho}_{0}+\bar{\rho}_{1}\ln\theta_{T}=\rho_{0}+\rho_{1}\ln\theta_{T}$

∎

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# Gravitational radiation pulses from Extreme-Mass-Ratio-Inspiral system with
a supermassive boson star

Yu-Peng Zhang1<EMAIL_ADDRESS>Yan-Bo Zeng1<EMAIL_ADDRESS>Yong-Qiang Wang1<EMAIL_ADDRESS>Shao-Wen Wei1<EMAIL_ADDRESS>Pau Amaro Seoane3, 4, 5<EMAIL_ADDRESS>Yu-Xiao Liu1<EMAIL_ADDRESS>corresponding author 1Lanzhou Center for Theoretical Physics, Key Laboratory
of Theoretical Physics of Gansu Province, School of Physical Science and
Technology, Lanzhou University, Lanzhou 730000, China
2Institute of Theoretical Physics & Research Center of Gravitation, Lanzhou
University, Lanzhou 730000, China
3Institute of Multidisciplinary Mathematics, Universitat Politècnica de
València, Spain
4DESY Zeuthen, Germany
5Kavli Institute for Astronomy and Astrophysics at Peking University, Beijing
100871, China
6Institute of Applied Mathematics, Academy of Mathematics and Systems Science,
CAS, Beijing 100190, China

###### Abstract

Future space-borne gravitational-wave detectors will observe the gravitational
waves in the milli-Hz. Extreme-mass-ratio inspirals with central supermassive
black holes are very important sources that could provide the information of
the vicinity of black holes. The event horizon separates the inner region of a
black hole and there is nothing that can escape from this region. When the
central supermassive compact object is a regular and horizonless rotating
boson star, a small body could pass through the center and follow novel types
of orbits. These will generate the gravitational waves that can not be
obtained in the scenario corresponding to an extreme-mass-ratio inspiral with
a central supermassive black hole. This can be used to examine whether a
supermassive rotating boson star is present at the centers of galaxies. In
this work, we consider an extreme-mass-ratio inspiral system described by a
small compact object inspiralling into a central supermassive rotating boson
star. Integrating four types of special equatorial geodesics and using the
numerical kludge method with quadrupole approximation, we obtain the
corresponding gravitational waveforms and find that there are high-frequency
gravitational radiation pulses in such system. The frequencies of the
gravitational radiation pulses could be in the magnitude of $10^{-1}$Hz and
the whole gravitational wave parts are in the milli-Hz. These novel
gravitational waveforms are quite different from that produced by the extreme-
mass-ratio inspiral system with a supermassive central black hole. By assuming
the masses of the central supermassive rotating boson star and small compact
object to be $10^{6}M_{\odot}$ and $10M_{\odot}$ and assuming a distance of
$1\text{Gpc}$, we show that the gravitational radiation pulses could be
detected by the space-borne gravitational-wave detectors. Our results will
provide a possible evidence to distinguish the astrophysical compact objects
in the galactic centers.

##  I Introduction

In 2015, LIGO successfully detected the gravitational waves (GWs) generated by
the merger of binary black holes Abbott2016a , which declares that the era of
gravitational wave astronomy is coming. Besides LIGO and VIRGO, there are
several independent space-borne detectors on their way, such as the Laser
Interferometer Space Antenna lisapaper ; lisawebsite , Taiji taiji , Tianqin
Luo2016 , and DECIGO Shuichi2017 . They might be launched into the space in
the next ten years and the detectable frequency of GWs will be extended into
the milli-Hz. Extreme-mass-ratio-inspirals (EMRIs) Amaro-Seoane2018b  are the
one type of most promise sources with the frequency in milli-Hz. They are
described by the late capture orbits of stellar-mass compact objects
inspiralling into central supermassive black holes in galactic centers. The
corresponding GWs radiated from such systems are closely related to the
accurate information in the vicinity of the central objects. It is well known
that the existence of event horizon for a black hole is one of the most
intriguing predictions of general relativity, which means there is nothing can
escape from the inside of the event horizon. Thus, it can be considered as the
merger of an EMRI system when the small star plunges into the event horizon of
the central supermassive black hole. The final stage of the GWs will have
gravitational radiation pulses Davis:1972ud  when the stellar-mass compact
object plunges into the central supermassive black hole.

Boson stars as the black hole mimickers had been proposed in the 1970s
Feinblum1968 ; Kaup1968 ; Ruffini1969 , they are formed by the gravitationally
bound Bose-Einstein condensates for light scalars with long de Broglie
wavelengths. Boson stars could have spherical Seidel1994 ; Giovanni2018  or
rotating Sanchis2019  configurations. Combining the observational data and the
possible stellar distribution in the background of a central boson star, the
possibility that whether a supermassive boson star can be in the center of a
galaxy was discussed in Ref. diego2000 . Later, the work of Amaro-
SeoaneBarrancoBernalRezzolla10  exploits the orbits of the S-stars at the
Galactic Centre to impose constraints to a hypothetical supermassive boson
star. What are the possible observations that could be seen with a central
supermassive boson star is still investigated Yuan:2004sv . By comparing the
power spectrum of a simple accretion disk model between a spherical boson star
and a Schwarzchild black hole, the result of Ref. Rueda2009  shows that it is
possible to find a boson star mimics the power spectrum of the disk of a black
hole with the same mass. The shadow of a boson star still provides the
possibility that a spherical Proca star could mimic a Schwarzchild black hole
Herdeiro2021lwl . Therefore, it is necessary to study what is the possible GWs
from the EMRIs with central supermassive rotating boson stars.

As the space-borne gravitational-wave detection program continues to advance,
the detection of the GWs from EMRIs is becoming possible. Boson star is a
regular and horizonless strong gravity system, which means it is particular
well motivated for considering a supermassive boson star located in the center
of a galaxy. The horizonless property means that a small star could plunge
into and pass through the center of a boson star. That is to say, the GWs
generated by such system in the final stage will be quite different from the
EMRIs with central supermassive black holes. Reference Kesden2005  has proven
that the novel GWs will exist in the central supermassive spherical boson
star. Note that, for a rotating boson star, the novel geodesics that are
different from a Kerr black hole have been obtained philippe2014 ; Grould12017
; Collodel2018 ; Yuzhang2021 . With such novel geodesic orbits, the
corresponding GWs will be really critical to study the properties of boson
stars. In this work, we will use these kind of orbits to obtain the
corresponding possible novel GWs.

To describe the gravitational mass scale of various boson stars, a natural
length scale (the reduced Compton wavelength)

$\bar{\lambda}=\frac{\hbar}{\mu c}$ (1)

is necessary, where $\mu$ is the scalar field mass parameter, $c$ is the speed
of light. The order of the magnitude of the boson star mass related with
$\bar{\lambda}$ is Palenzuela2012

$M_{\text{bos}}\sim\frac{c^{2}~{}\bar{\lambda}}{G}=\frac{m^{2}_{p}}{\mu},$ (2)

where the Planck mass $m_{p}=\sqrt{\hbar c/G}\approx 2.18\times
10^{-8}~{}\text{kg}$. For a boson star with ultra-light mass parameter, the
mass scale could be comparable with a supermassive black hole Palenzuela2012 ;
philippe2014 .

In this work, we consider the EMRI system described by a stellar-mass compact
object inspiralling into a central supermassive rotating boson star and
investigate the corresponding GWs with the novel orbits. If the supermassive
compact object located in the center of a galaxy is a boson star, the EMRI
system will be different and what are the possible observable GWs will be very
important. We will use the numerical kludge (NK) method Stanislav2007  to
obtain the approximate GWs emitted by a small object inspiralling into a
central supermassive boson star.

##  II Equatorial geodesics and Gravitational waves

In order to construct the equilibrium configurations of rotating boson stars,
we take the following ansatz for the metric Herdeiro2014 ; Herdeiro2015

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle
g_{tt}dt^{2}+g_{rr}dr^{2}+g_{\theta\theta}d\theta^{2}+2g_{t\varphi}dtd\varphi+g_{\varphi\varphi}d\varphi^{2}$
(3) $\displaystyle=$
$\displaystyle-e^{2F_{0}}dt^{2}+e^{2F_{1}}\left(dr^{2}+r^{2}d\theta^{2}\right)$
$\displaystyle+~{}e^{2F_{2}}r^{2}\sin^{2}\theta(d\varphi-Wdt)^{2}.$

We consider the mini boson star Schunck2003  with following action

$S=\int{d^{4}x\sqrt{-g}\left[\frac{R}{16\pi
G}-\nabla_{\mu}\Phi\nabla^{\mu}\Phi^{*}-\frac{\mu^{2}}{\hbar^{2}}\Phi\Phi^{*}\right]},$
(4)

where $G$ is Newton’s gravitational constant and $m$ is the mass parameter of
the scalar field. We set $G$, the speed of light $c$, and the Planck constant
$\hbar$ to be unity ($G=c=\hbar=1$). For a stationary and axisymmetric
rotating boson star, the ansatz for the complex scalar field $\Phi$ is

$\Phi=\phi(r,\theta)\exp\left[i(\omega t-k\varphi)\right],$ (5)

where $\omega$ is the frequency of the scalar field. The configuration of a
equilibrium boson star could be obtained with the suitable boundary
conditions, see the details in Ref. Herdeiro2015 . Here, we only consider a
rotating boson star with $k=1$ and $\omega=0.79m$.

A rotating boson star is stationary and axisymmetry, which means there are a
timelike killing vector $\xi^{\mu}=(\partial_{t})^{\mu}$ and a spacelike
killing vector $\eta^{\mu}=(\partial_{\varphi})^{\mu}$. These two independent
Killing vectors lead to the following two conserved quantities:

$\displaystyle\bar{E}$ $\displaystyle=$
$\displaystyle-(\partial_{t})^{\mu}u_{\mu}=-g_{tt}u^{t}-g_{t\varphi}u^{\varphi},$
(6) $\displaystyle\bar{J}$ $\displaystyle=$
$\displaystyle(\partial_{\varphi})^{\mu}u_{\mu}=g_{t\varphi}u^{t}+g_{\varphi\varphi}u^{\varphi},$
(7)

where $\bar{E}$ and $\bar{J}$ are the energy and angular momentum of the
particle per unit mass. For simplicity, we only consider a test particle that
moves in the equatorial orbits, for which the four-velocity is

$u^{\mu}=\left(\frac{dt}{d\tau},\frac{dr}{d\tau},0,\frac{d\varphi}{d\tau}\right)=\left(u^{t},u^{r},0,u^{\varphi}\right),$
(8)

where $\tau$ is the proper time of the test particle. Solving Eqs. (6)-(7) and
$u^{\mu}u_{\mu}=-1$, we have the following non-vanishing components of the
velocity

$\displaystyle u^{t}$ $\displaystyle=$
$\displaystyle\frac{\bar{E}g_{\varphi\varphi}+\bar{J}g_{t\varphi}}{g_{t\varphi}^{2}-g_{tt}g_{\varphi\varphi}},$
(9) $\displaystyle u^{\varphi}$ $\displaystyle=$
$\displaystyle\frac{\bar{E}g_{t\varphi}+\bar{J}g_{tt}}{g_{tt}g_{\varphi\varphi}-g_{t\varphi}^{2}},$
(10) $\displaystyle\left(u^{r}\right)^{2}$ $\displaystyle=$
$\displaystyle-\frac{1+g_{\varphi\varphi}u^{\varphi}u^{\varphi}+2g_{t\varphi}u^{t}u^{\phi}+g_{tt}u^{t}u^{t}}{g_{rr}}.$
(11)

Using the effective potential is an easy way to determine the radial motion of
a test particle in a central field. By decomposing the form of Eq. (11) as

$\displaystyle(u^{r})^{2}\propto\left(\bar{E}-V^{+}_{\text{eff}}\right)\left(\bar{E}-V^{-}_{\text{eff}}\right)$
(12)

we can obtain the effective potentials $V^{+}_{\text{eff}}$ and
$V^{-}_{\text{eff}}$ of a test particle in the background of a rotating boson
star Grould12017 ; Collodel2018 ; Yuzhang2021 . By checking the relation
between the effective potentials and angular momentum $\bar{J}$, one can prove
that a test particle could be in a circular orbit with arbitrary angular
momentum and there is no innermost stable circular orbit for the test particle
in a rotating boson star. This special property leads to that the velocity of
the test particle in the rotating boson star satisfies

$(u^{r},u^{\varphi})=(0,0)$ (13)

at some special points where the angular momentum and energy satisfy the
following condition Collodel2018

$\bar{E}=-\frac{\bar{J}g_{tt}}{g_{t\varphi}}.$ (14)

Combining the effective potentials (12) and (14) and choosing suitable energy
and angular momentum for the small body, we can obtain the novel orbits that
do not appear in the Kerr black hole case.

In this work, we consider four types of special orbits in a rotating boson
star. These novel orbits can be used to distinguish boson stars from black
holes. We give the periastron $r_{p}$, apastron $r_{a}$, angular momentum
$\bar{J}$, and energy $\bar{E}$ of these orbits in Table 1. With the help of
the definition of the orbital eccentricity

$e=\frac{r_{a}-r_{p}}{r_{a}+r_{p}},$ (15)

it is easy to prove that the orbital eccentricity $e$ for a test particle in a
rotating boson star could be $1$, this result is unacceptable for a black
hole.

Figure 1 shows the corresponding orbits that we considered. For the orbit (a),
the periastron $r_{p}$ is less than the corresponding effective Schwarzchild
radius of the boson star. For the orbit (b), the test particle could pass
through the center of the rotating boson star and the periastron $r_{p}=0$.
For the orbit (c), the radial velocity and angular velocity will be zero when
the test particle is located at the periastron. For the orbit (d), the radial
velocity and angular velocity will be zero when the test particle is located
at the apastron. The orbits (a), (b), and (d) have the same apastrons but
different periastrons. These four of special orbits will offer possible GW
signals to check whether spuermassive boson stars exist.

Table 1: Values of the parameters for the four types of orbits. orbit |  $~{}\bar{J}~{}$ |  $r_{a}/M$ |  $r_{p}/M$ |  $\bar{E}$ |  $e$
---|---|---|---|---|---
a | 0.15 | 9.41476 | 0.10650 | 0.86788 | 0.97735
b | 0 | 9.41476 | 0 | 0.86746 | 1
c | -0.15 | 2.07405 | 0.92260 | 0.61943 | 0.38426
d | -0.30 | 9.41476 | 0.20175 | 0.86713 | 0.95777
Figure 1: Four special types of orbits of a small body in a rotating boson
star.

Next, we compute the corresponding GWs of the stellar-mass compact object
inspiralling into a supermassive rotating boson star with the orbits given in
Fig. 1. In general, the GWs from an EMRI can be determined accurately by using
the perturbation theory Teukolsky1973 ; Hughes:2001jr . For a rotating boson
star, the background solution is numerical and it is really hard to obtain the
gravitational perturbation equations. Therefore, we use the numerical kludge
(NK) method Stanislav2007  to generate the corresponding GWs for the EMRI
system with a central supermassive rotating boson star.

Using the well-known solutions of the linearized gravitational perturbation
and combining the quadrupole formula in terms of the geodesic trajectory, we
could get the approximate GWs. By integrating the four-velocity, we can obtain
the corresponding trajectories. Once the trajectories of the small body have
been obtained, one can construct the effective trajectories by projecting the
Boyer-Lindquist coordinates into a pseudo-flat space as follows

$x=r\sin\theta\cos\varphi,~{}y=r\sin\theta\sin\varphi,~{}z=r\cos\theta.$ (16)

Then we can obtain the GWs with the help of the geodesic trajectories and the
“quadrupole approximation”

$\bar{h}^{ij}=\frac{2G}{d}\ddot{I}^{jk}|_{t^{\prime}=r-d},$ (17)

where the parameter $d$ is the distance from the source to the observation
point. The definition of the quadrupole $I^{jk}$ is

$I^{jk}(t^{\prime})=\int{x^{\prime j}x^{\prime
k}T^{00}(t^{\prime},x^{\prime})d^{3}x^{\prime}},$ (18)

where the coordinate $x^{\prime j}$ is the position of a small star along the
corresponding geodesic trajectory described by (16). We compute the GWs of the
cross and plus polarizations by adopting the Transverse-Traceless projection
result (17) based on Ref. Kesden2005 .

The calculation of the GWs is done with the NK method. With such
approximation, the corresponding amplitudes of the GWs will be inaccurate.
Therefore, our calculation is “phenomenological” but the frequencies of the
GWs are correct. Reference Kesden2005  showed that the kludge waveforms are
extremely well at approximating the true gravitational waveform and they have
overlaps with the Teukolsky waveforms of $95\%$ Kesden2005 . That is to say,
it is possible to use these kludge waveforms with achievable quality to
provide some roles in the search of space-borne gravitational-wave data for
EMRIs with central supermassive rotating boson stars.

In principle, the information of the waveforms should show the properties of
the corresponding orbits. For the orbits obtained in Fig. 1, we expect that we
could observe novel GWs, especially for the orbits (c) and (d). Figure 2 shows
the time-domain GWs generated by the orbits in Fig. 1 viewed at an inclination
angle $90^{\circ}$, where the results in right panel (a2, b2, c2, d2)
correspond to the zooms of the GWs in left panel (a1, b1, c1, d1). Checking
the corresponding GWs in Fig. 2, we find that there are quasi gravitational
radiation pulses generated by the orbits (a) and (d), and it is absent for the
orbits (b) and (c).

Figure 2: Amplitudes of the plus polarization for the GWs generated by the
orbits in Fig. 1.

In view of the orbits in Fig. 1 and the GWs in Fig. 2, the dependence of the
corresponding GWs and orbits does not seem to be the same as our expectation.
By analyzing the corresponding velocities and accelerations at different
positions along the orbits of the test particle, we find that although there
are peaks in the orbits (c) and (d), only the orbit (d) can generate
gravitational radiation pulses. In addition, the orbit (a) can also generate
gravitational radiation pulses. To get the relations between the GWs and
orbits, we go back to the definition of the waveform (17). Following Eq. (17)
we know that the amplitudes of the GWs are proportional to the second-order
time derivative of the quadrupole, which means the GWs are dependent on the
orbit location, velocity, and acceleration of the test particle. Note that the
orbital eccentricities of the orbits (a) and (d) are close to but not equal to
$1$, the corresponding velocity $(u^{r},u^{\varphi})$ will change greatly in a
very short time when the test particle is passing through the periastron
$r_{p}$. This change is the main cause of gravitational radiation pulses.
Figure 3 shows the velocity $(u^{r},u^{\varphi})$ and acceleration
$(du^{r}/d\tau,du^{\varphi}/d\tau)$ of each orbit given in Fig. 1. These
behaviors are not possessed by the supermassive black holes. Combined with the
future space-borne gravitational wave detection plan, it could be possible to
verify whether there are supermassive boson stars in galactic centers.

Figure 3: Velocity $(u^{r},u^{\varphi})$ and acceleration
$(du^{r}/d\tau,du^{\varphi}/d\tau)$ of the orbits in Fig. 1. The results in
subfigures (i1-i4) correspond to the orbit (i), where $i=(a,b,c,d)$.

Next, we discuss the detectability of the GW signals we obtained. The
corresponding mass of the rotating boson star is

$M\approx 1.30\frac{m_{p}^{2}}{\mu}.$ (19)

Here, we should note that, the prediction of Ref. Amaro-Seoane:2010pks  showed
that the mass parameter of the scalar field is of about
$\mu\in[300\,\text{eV},\,2\times 10^{4}\,\text{eV}]$. This prediction is
obtained in terms of the astronomical observations of Sgr $A^{*}$, NGC 4258,
and Collisional dark matter. However, it is shown that the total number of
galaxies in the univesrse up to $z=8$ is almost $10^{12}$ Christopher2016 .
Therefore, faced with such a large number of galaxies, we cannot strictly rule
out the absence of supermassive boson stars in the galactic centers. If we set
the mass parameter of the scalar field to be $\mu\approx 10^{-16}\text{eV}$,
then the mass of the rotating boson star can be $M=10^{6}M_{\odot}$.

We set the mass of the small star to be $m=10M_{\odot}$. The distance from the
detectors to the source is set up to $1\text{Gpc}$. With such assumptions, we
could estimate the maximal amplitudes and frequencies of the GWs. We find that
the frequencies of the pulse part could be in the magnitude of $10^{-1}$Hz,
while the frequencies of the whole GW signals are still in the milli-Hz. By
naively comparing the LISA’s sensitivity window and the GW signals in Fig. 4,
we find that these novel GW signals could be detected by the space-borne
gravitational-wave detectors.

Figure 4: Characteristic amplitude of the GW signals given in Fig. 2 and
LISA’s sensitivity window Moore2015 , where the points are determined by the
maximal amplitudes and frequencies of the whole GW signals, (a-d) indicates
the corresponding orbits of the GW signals.

##  III Conclusions

In this work, we obtain the GWs with the quadrupole approximation for an EMRI
system with a central supermassive rotating boson star. We consider four types
of special orbits (Fig. 1) and derive the corresponding GWs. Due the
horizonless property of the rotating boson star, the stellar-mass compact
object could pass through the center of the rotating boson star and the
orbital eccentricity could be $e=1$. The distribution of the scalar field in
the whole space also allows the stellar-mass compact object to be at rest for
some special orbits (the orbits (c) and (d) in Fig. 1). We calculate the
corresponding GWs from the orbits in Fig. 1 and find that there are
gravitational radiation pulses when the stellar-mass compact object moves
along the orbits (a) and (d). Although the orbits (b) and (c) have peaks,
there are no gravitational radiation pulses generated by them. These novel GWs
are quite different from the EMRI system with a central supermassive Kerr
black hole, and can therefore be envisaged as a way to discover supermassive
rotating boson stars. Assuming for the boson star and stellar compact object
masses $M=10^{6}M_{\odot}$ and $m\approx 10M_{\odot}$, approximately, and
assuming a distance of $d=1{\text{Gpc}}$ between the source and detector we
showed that such system could be detected by the future space-borne
gravitational-wave detectors.

Although our numerical method is not accurate enough compared with the
perturbation method, the research provides a stepping stone for the study of
supermassive boson stars in galaxies with the help of the future space
gravitational-wave detection.

##  IV Acknowledgments

This work was supported in part by the National Key Research and Development
Program of China (Grant No. 2020YFC2201503), the National Natural Science
Foundation of China (Grants No. 11875151, No. 11705070, No. 12105126, and No.
12075103), the 111 Project under (Grant No. B20063), the Fundamental Research
Funds for the Central Universities (Grants No. lzujbky-2021-pd08 and
lzujbky-2019-ct06), and “Lanzhou City’s scientific research funding subsidy to
Lanzhou University”. PAS acknowledges support from the Ramón y Cajal Programme
of the Ministry of Economy, Industry and Competitiveness of Spain, as well as
the financial support of Programa Estatal de Generación de Conocimiento (ref.
PGC2018-096663-B-C43) (MCIU/FEDER). This work was supported by the National
Key R&D Program of China (2016YFA0400702) and the National Science Foundation
of China (11721303).

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