105 Q&A from Bayesian Forecasting and Dynamic Models - Chapter 4

105.1 Questions from Chapter 4

Tip

Unless explicitly stated, the questions below assume a standard, univariate DLM with known variances, i.e., \{F_t , G_t , V_t , W_t \} defined by

Y_t = F_t \theta_t + \nu_t, \qquad \nu_t \sim \mathcal{N}[0, V_t],

\theta_t = G_t \theta_{t-1} + \omega_t, \qquad \omega_t \sim \mathcal{N}[0, W_t],

(\theta_{t-1} \mid \mathcal{D}_{t-1}) \sim \mathcal{N}[m_{t-1}, C_{t-1}].

Exercise 105.1 Consider the DLM \{F_t , G, V_t , W_t \}.

If G is of full rank, prove that the DLM can be reparametrized to the DLM

\left\{ \begin{pmatrix} F_t & 0 \end{pmatrix}, \begin{pmatrix} G & 0 \\ 1 & 0 \end{pmatrix}, V_t, \begin{pmatrix} W_t & 0 \\ 0 & 0 \end{pmatrix} \right\}.

Now show how to accommodate the observation variance in the system equation when G is singular.

Exercise 105.2 Consider the constant DLM \{F, G, V, W\}, generalised so that

\nu_t \sim \mathcal{N}[\bar{v}, V]

\omega_t \sim \mathcal{N}[\bar{w}, W].

and

Show that (\theta_t \mid \mathcal{D}_t) \sim \mathcal{N}[m_t, C_t] and derive recurrence relationships for m_t and C_t

by using Bayes’ theorem for updating, and
by deriving the joint distribution of (\theta_t , Y_t \mid \mathcal{D}_{t-1}) and using normal theory to obtain the conditional probability distribution.
What is the forecast function f_t (k)?
How do the results generalise to the DLM \{F, G, V, W\}_t with

\nu_t \sim \mathcal{N}[\bar{v}_t , V_t]

and

\omega_t \sim \mathcal{N}[\bar{w}_t , W_t]?

Exercise 105.3

For the constant DLM \{F, G, V, W\}, given (\theta_t \mid \mathcal{D}_t) \sim \mathcal{N}[m_t, C_t], obtain

the k-step ahead forecast distribution p(Y_{t+k} \mid \mathcal{D}_t);
the k-step lead-time forecast distribution p(X_{t,k} \mid \mathcal{D}_t) where

X_{t,k} = \sum_{r=1}^{k} Y_{t+r}.

Exercise 105.4 For the univariate DLM define b_t = V_t / Q_t.

Show that given \mathcal{D}_t, the posterior distribution for the mean response \mu_t = F_t' \theta_t is (\mu_t \mid \mathcal{D}_t) \sim \mathcal{N}[f_t (0), Q_t (0)], where E[\mu_t \mid \mathcal{D}_t] = f_t (0) = F_t' m_t and V[\mu_t \mid \mathcal{D}_t] = Q_t (0) = F_t' C_t F_t. Use the recurrence equations for m_t and C_t to show that for some appropriate scalar A_t that you should define,
E[\mu_t \mid \mathcal{D}_t] can be updated using either the equation E[\mu_t \mid \mathcal{D}_t] = E[\mu_t \mid \mathcal{D}_{t-1}] + A_t e_t or f_t (0) = A_t Y_t + (1 - A_t) f_t, interpreting f_t (0) as a weighted average of two estimates of \mu_t;
V[\mu_t \mid \mathcal{D}_t] can be updated using either the equation V[\mu_t \mid \mathcal{D}_t] = (1 - A_t) V[\mu_t \mid \mathcal{D}_{t-1}] or V[\mu_t \mid \mathcal{D}_t] = Q_t (0) = A_t V_t.

Exercise 105.5 Write H_t = W_t - W_t F_t F_t' W_t / Q_t, L_t = (1 - A_t) W_t F_t, and A_t = 1 - V_t / Q_t. Prove that the posterior distribution of the observation and evolution errors is

\begin{pmatrix} \nu_t \\ \omega_t \end{pmatrix} \Bigm| \mathcal{D}_t \sim \mathcal{N} \!\left[ \begin{pmatrix} (1 - A_t)\,e_t \\ -\,L_t\,e_t \end{pmatrix}, \begin{pmatrix} (1 - A_t)\,V_t & -\,L_t' \\ -\,L_t & H_t \end{pmatrix} \right].

Exercise 105.6 Consider the DLM \{F_t , G_t , V_t , V_t W_t^\ast\} with unknown variances V_t, but in which the observational errors are heteroscedastic, so that

\nu_t \sim \mathcal{N}[0, k_t V],

where V = \phi^{-1} and k_t is a known, positive variance multiplier. Also,

(\phi \mid \mathcal{D}_{t-1}) \sim \mathcal{G}\big[n_{t-1}/2, \; n_{t-1} S_{t-1}/2\big].

What is the posterior (\phi \mid \mathcal{D}_t)?
How are the summary results of Section 4.6 affected?

Exercise 105.7 Consider the closed, constant DLM \{1, \lambda, V, W\} with \lambda, V and W known, and |\lambda| < 1.

Obtain the k-step forecast distribution (Y_{t+k} \mid \mathcal{D}_t) as a function of m_t , C_t and \lambda.
Show that as k \to \infty, (Y_{t+k} \mid \mathcal{D}_t) converges in distribution to \mathcal{N}\!\big[0, V + W/(1 - \lambda^{2})\big].
Obtain the joint forecast distribution for (Y_{t+1} , Y_{t+2} , Y_{t+3}).
Obtain the k-step lead-time forecast distribution p(X_{t,k} \mid \mathcal{D}_t) where

X_{t,k} = \sum_{r=1}^{k} Y_{t+r}.

Exercise 105.8

Generalise Theorem 4.1 to a DLM whose observation and evolu- tion noise are instantaneously correlated. Speciﬁcally, suppose that C[\omega_t , \nu_t ] = c_t, a known n-vector of covariance terms, but that all other assumptions remain valid. Show now that Theorem 4.1 applies with the modiﬁcations

Q_t = (F_t' R_t F_t + V_t) + 2 F_t' c_t

and

A_t = (R_t F_t + c_t)/Q_t .

If G is of full rank, show how to reformulate the DLM in the standard form so the observation and evolution noise are uncorrelated.

Exercise 105.9 Given \mathcal{D}_t, your posterior distribution is such that

(\theta_t \mid \mathcal{D}_t) \sim \mathcal{N}[m_t, C_t]

and

(\theta_{t-k} \mid \mathcal{D}_t) \sim \mathcal{N}[a_t(-k), R_t(-k)].

The regression matrix of \theta_{t-k} on \theta_t is A_{t-k,t}. You are now about to receive additional information Z, that might be an external forecast, expert opinion, or more observations. If

\perp Z \mid \theta_t,

(\theta_1 , \ldots , \theta_{t-1}) \perp

and given the information Z, your revised distribution is

(\theta_t \mid \mathcal{D}_t , Z) \sim \mathcal{N}[m_t + \Delta , C_t - \Sigma],

what is your revised distribution for (\theta_{t-k} \mid \mathcal{D}_t , Z)?

Exercise 105.10 Prove the retrospective results, sets (i) and (ii) of Theorem 4.5, using the conditional independence structure of the DLM and the conditional independence results of the Appendix, Section 4.11.

Exercise 105.11 With discount factor \delta, the discount regression DLM \{F_t , I, V, W_t \} is such that I is the identity matrix and W_t = C_{t-1} (1-\delta)/\delta. Given \mathcal{D}_t, and with integer k > 0, show that

R_t = C_{t-1} / \delta;
the regression matrix of \theta_{t-k} on \theta_{t-k+1} is B_{t-k} = \delta I;
the regression matrix of \theta_{t-k} on \theta_t is A_{t-k,t} = \delta^{k} I;
the filtering recurrences of Theorem 4.5 simplify to
a_t (-k) = a_{t-1} (-k + 1) + \delta^{k} A_t e_t, R_t (-k) = R_{t-1} (-k + 1) - \delta^{2k} A_t Q_t A_t',

a_t (-k) = m_{t-k} + \delta[a_t (-k + 1) - a_{t-k+1}], R_t (-k) = C_{t-k} + \delta^{2} [R_t (-k + 1) - R_{t-k+1}].

Exercise 105.12 Generalise Theorem 4.1 to the multivariate DLM \{F_t , G_t , V_t , W_t \}.

Exercise 105.13 Verify the filtering and smoothing results in Corollaries 4.3 and 4.4 relating to the DLM with an unknown constant observational variance V .

Exercise 105.14 Prove the reference prior results stated in part (1) of Theorem 4.12, that relate to the case of a known observational variance V .

Exercise 105.15 (Reference analysis updating for the first-order polynomial model) Prove the reference prior results stated in part (2) of Theorem 4.13, relating to the case of an unknown observational variance V.

Exercise 105.16 (Reference analysis updating for the first-order polynomial model) Prove the results stated in Corollary 4.8, providing the conversion from reference analysis updating to the usual recurrence equations when posteriors become proper.

Exercise 105.17 (Reference analysis updating for the first-order polynomial model) Consider the first-order, polynomial model \{1, 1, V, W\}, with n = 1 and \theta_t = \mu_t, in the reference analysis updating of Section 4.10.

Assume that V is known. Using the known variance results of Theorem 4.12 and Corollary 4.8, show that (\mu_1 \mid \mathcal{D}_1) \sim \mathcal{N}[Y_1, V].
Assume that V = 1/\phi is unknown, so that the results of Theorem 4.13 and Corollary 4.8 apply. Show that the posterior for \mu_t and V becomes proper and of standard form at t = 2, and identify the defining quantities m_2, C_2, n_2 and S_2.
In (b), show directly how the results simplify in the case W = 0.

Exercise 105.18 (Reference analysis updating for the 2-dimensional model) Consider the 2-dimensional model

\left\{ \begin{pmatrix} 1 & 0 \end{pmatrix}, \begin{pmatrix} 1 & 1 \\[4pt] 0 & 1 \end{pmatrix}, V, 0 \right\}

assuming V to be known. Apply Theorem 4.13 and Corollary 4.8 to deduce the posterior for (\theta_2 \mid \mathcal{D}_2).

Exercise 105.19 (Discount regression DLM with unknown variance) Consider the discount DLM \{F_t , I, V, V W_t^*\} with unknown but constant variance V. The discount factor is 0 < \delta \le 1, so that W_t^* = C_{t-1}^* (1 - \delta)/\delta, and R_t^{*-1} = \delta C_{t-1}^{*-1}. Initially, ignorance is formulated so the prior precision of \theta_0 is C_0^{-1} = 0, the prior degrees of freedom for \phi = 1/V are n_0 = 0, m_0 = 0 and S_0 = 0. Use the notation of Theorem 4.3 so that starred variances are conditioned on V = 1.

Prove that C_1^{*-1} = F_1 F_1'.
Prove that

C_t^{*-1} = \sum_{v=0}^{t-1} \delta^{v}\, F_{t-v} F_{t-v}'.

Suppose that \operatorname{rank}\, C_t^{*-1} = \operatorname{rank}\, C_{t-1}^{*-1} = r_{t-1}. Show that the relationship between Y_t and the information \mathcal{D}_{t-1} can be modelled by a DLM \{\tilde{F}_t , I, V, V \tilde{W}_t \} with a parameter of dimension r_{t-1} that, conditional upon V, has a proper distribution. Con- sequently, show that a conditional forecast exists such that (Y_t \mid \mathcal{D}_{t-1}, V) \sim \mathcal{N}[f_t , V Q_t^*], and that given Y_t, the variance can be updated according to n_t = n_{t-1} + 1, S_t = S_{t-1} + (e_t^{2}/Q_t^* - S_{t-1})/n_t, with n_1 = 1 and S_1 = e_1^{2}. Show further that if n_{t-1} > 0, the unconditional forecast (Y_t \mid \mathcal{D}_{t-1}) \sim \mathcal{T}_{n_{t-1}}[f_t , Q_t] exists. Suggest a method of obtaining m_t for this reduced DLM.
Now suppose \operatorname{rank}\, C_t^{*-1} = 1 + \operatorname{rank}\, C_{t-1}^{*-1}. Show that no forecast of (Y_t \mid \mathcal{D}_{t-1}, V) is possible. However, given Y_t, show that although \{n_t , S_t \} = \{n_{t-1} , S_{t-1} \}, the dimension of the design space for which forecasts can now be derived is increased by 1.
Collinearity can be a real problem. For example, in the simplest regression discount DLM, a price variable used as a regressor may be held constant for quite a long period before being changed. Obviously, information is being gathered, so that useful conditional forecasts can be made based upon this price. However, forecasts conditional on other prices cannot be made until after a price change has been experienced. In general, at time t - 1, the forecast design space is spanned by F_1 , \ldots , F_{t-1}. Construct an algorithm to provide a reference analysis accommodating collinearity that at time t - 1, enables forecasts and monitoring for time t, whenever F_t is contained in the current forecast design space and n_{t-1} \ge 1.
Generalise this approach beyond discount regression DLMs. Some relevant work is presented in Vasconcellos (1992).

--- title: "Q&A from Bayesian Forecasting and Dynamic Models - Chapter 4" --- ## Questions from Chapter 4 ::: {.callout-tip} Unless explicitly stated, the questions below assume a standard, univariate DLM with known variances, i.e., $\{F_t , G_t , V_t , W_t \}$ defined by $$ Y_t = F_t \theta_t + \nu_t, \qquad \nu_t \sim \mathcal{N}[0, V_t], $$ $$ \theta_t = G_t \theta_{t-1} + \omega_t, \qquad \omega_t \sim \mathcal{N}[0, W_t], $$ $$ (\theta_{t-1} \mid \mathcal{D}_{t-1}) \sim \mathcal{N}[m_{t-1}, C_{t-1}]. $$ ::: ::: {#exr-ch4-ex1} Consider the DLM $\{F_t , G, V_t , W_t \}$. (a) If $G$ is of [full rank](https://en.wikipedia.org/wiki/Rank_(linear_algebra)), prove that the DLM can be reparametrized to the DLM $$ \left\{ \begin{pmatrix} F_t & 0 \end{pmatrix}, \begin{pmatrix} G & 0 \\ 1 & 0 \end{pmatrix}, V_t, \begin{pmatrix} W_t & 0 \\ 0 & 0 \end{pmatrix} \right\}. $$ (b) Now show how to accommodate the observation variance in the system equation when $G$ is singular. :::: ::: {#exr-ch4-ex2} Consider the constant DLM $\{F, G, V, W\}$, generalised so that $$ \nu_t \sim \mathcal{N}[\bar{v}, V] $$ $$ \omega_t \sim \mathcal{N}[\bar{w}, W]. $$ and Show that $(\theta_t \mid \mathcal{D}_t) \sim \mathcal{N}[m_t, C_t]$ and derive recurrence relationships for $m_t$ and $C_t$ (a) by using Bayes’ theorem for updating, and (b) by deriving the joint distribution of $(\theta_t , Y_t \mid \mathcal{D}_{t-1})$ and using normal theory to obtain the conditional probability distribution. (c) What is the forecast function $f_t (k)$? (d) How do the results generalise to the DLM $\{F, G, V, W\}_t$ with $$ \nu_t \sim \mathcal{N}[\bar{v}_t , V_t] $$ and $$ \omega_t \sim \mathcal{N}[\bar{w}_t , W_t]? $$ :::: ::: {#exr-ch4-ex3} (3) For the constant DLM $\{F, G, V, W\}$, given $(\theta_t \mid \mathcal{D}_t) \sim \mathcal{N}[m_t, C_t]$, obtain (a) the $k$-step ahead forecast distribution $p(Y_{t+k} \mid \mathcal{D}_t)$; (b) the $k$-step lead-time forecast distribution $p(X_{t,k} \mid \mathcal{D}_t)$ where $$ X_{t,k} = \sum_{r=1}^{k} Y_{t+r}. $$ ::: ::: {#exr-ch4-ex4} For the univariate DLM define $b_t = V_t / Q_t$. (a) Show that given $\mathcal{D}_t$, the posterior distribution for the mean response $\mu_t = F_t' \theta_t$ is $(\mu_t \mid \mathcal{D}_t) \sim \mathcal{N}[f_t (0), Q_t (0)]$, where $E[\mu_t \mid \mathcal{D}_t] = f_t (0) = F_t' m_t$ and $V[\mu_t \mid \mathcal{D}_t] = Q_t (0) = F_t' C_t F_t$. Use the recurrence equations for $m_t$ and $C_t$ to show that for some appropriate scalar $A_t$ that you should define, (b) $E[\mu_t \mid \mathcal{D}_t]$ can be updated using either the equation $E[\mu_t \mid \mathcal{D}_t] = E[\mu_t \mid \mathcal{D}_{t-1}] + A_t e_t$ or $f_t (0) = A_t Y_t + (1 - A_t) f_t$, interpreting $f_t (0)$ as a weighted average of two estimates of $\mu_t$; (c) $V[\mu_t \mid \mathcal{D}_t]$ can be updated using either the equation $V[\mu_t \mid \mathcal{D}_t] = (1 - A_t) V[\mu_t \mid \mathcal{D}_{t-1}]$ or $V[\mu_t \mid \mathcal{D}_t] = Q_t (0) = A_t V_t$. ::: ::: {#exr-ch4-ex5} Write $H_t = W_t - W_t F_t F_t' W_t / Q_t$, $L_t = (1 - A_t) W_t F_t$, and $A_t = 1 - V_t / Q_t$. Prove that the posterior distribution of the observation and evolution errors is $$ \begin{pmatrix} \nu_t \\ \omega_t \end{pmatrix} \Bigm| \mathcal{D}_t \sim \mathcal{N} \!\left[ \begin{pmatrix} (1 - A_t)\,e_t \\ -\,L_t\,e_t \end{pmatrix}, \begin{pmatrix} (1 - A_t)\,V_t & -\,L_t' \\ -\,L_t & H_t \end{pmatrix} \right]. $$ ::: ::: {#exr-ch4-ex6} Consider the DLM $\{F_t , G_t , V_t , V_t W_t^\ast\}$ with unknown variances $V_t$, but in which the observational errors are heteroscedastic, so that $$ \nu_t \sim \mathcal{N}[0, k_t V], $$ where $V = \phi^{-1}$ and $k_t$ is a known, positive variance multiplier. Also, $$ (\phi \mid \mathcal{D}_{t-1}) \sim \mathcal{G}\big[n_{t-1}/2, \; n_{t-1} S_{t-1}/2\big]. $$ (a) What is the posterior $(\phi \mid \mathcal{D}_t)$? (b) How are the summary results of Section 4.6 affected? :::: ::: {#exr-ch4-ex7} Consider the closed, constant DLM $\{1, \lambda, V, W\}$ with $\lambda, V$ and $W$ known, and $|\lambda| < 1$. (a) Obtain the $k$-step forecast distribution $(Y_{t+k} \mid \mathcal{D}_t)$ as a function of $m_t , C_t$ and $\lambda$. (b) Show that as $k \to \infty$, $(Y_{t+k} \mid \mathcal{D}_t)$ converges in distribution to $\mathcal{N}\!\big[0, V + W/(1 - \lambda^{2})\big]$. (c) Obtain the joint forecast distribution for $(Y_{t+1} , Y_{t+2} , Y_{t+3})$. (d) Obtain the $k$-step lead-time forecast distribution $p(X_{t,k} \mid \mathcal{D}_t)$ where $$ X_{t,k} = \sum_{r=1}^{k} Y_{t+r}. $$ ::: ::: {#exr-ch4-ex8} (8) Generalise Theorem 4.1 to a DLM whose observation and evolu- tion noise are instantaneously correlated. Speciﬁcally, suppose that $C[\omega_t , \nu_t ] = c_t$, a known $n$-vector of covariance terms, but that all other assumptions remain valid. Show now that Theorem 4.1 applies with the modiﬁcations $$ Q_t = (F_t' R_t F_t + V_t) + 2 F_t' c_t $$ and $$ A_t = (R_t F_t + c_t)/Q_t . $$ If $G$ is of full rank, show how to reformulate the DLM in the standard form so the observation and evolution noise are uncorrelated. ::: :::: {#exr-ch4-ex9} Given $\mathcal{D}_t$, your posterior distribution is such that $$ (\theta_t \mid \mathcal{D}_t) \sim \mathcal{N}[m_t, C_t] $$ and $$ (\theta_{t-k} \mid \mathcal{D}_t) \sim \mathcal{N}[a_t(-k), R_t(-k)]. $$ The regression matrix of $\theta_{t-k}$ on $\theta_t$ is $A_{t-k,t}$. You are now about to receive additional information $Z$, that might be an external forecast, expert opinion, or more observations. If $$ \perp Z \mid \theta_t, $$ $$ (\theta_1 , \ldots , \theta_{t-1}) \perp $$ and given the information $Z$, your revised distribution is $$ (\theta_t \mid \mathcal{D}_t , Z) \sim \mathcal{N}[m_t + \Delta , C_t - \Sigma], $$ what is your revised distribution for $(\theta_{t-k} \mid \mathcal{D}_t , Z)$? :::: :::: {#exr-ch4-ex10} Prove the retrospective results, sets (i) and (ii) of Theorem 4.5, using the conditional independence structure of the DLM and the conditional independence results of the Appendix, Section 4.11. :::: :::: {#exr-ch4-ex11} With discount factor $\delta$, the discount regression DLM $\{F_t , I, V, W_t \}$ is such that $I$ is the identity matrix and $W_t = C_{t-1} (1-\delta)/\delta$. Given $\mathcal{D}_t$, and with integer $k > 0$, show that (a) $R_t = C_{t-1} / \delta$; (b) the regression matrix of $\theta_{t-k}$ on $\theta_{t-k+1}$ is $B_{t-k} = \delta I$; (c) the regression matrix of $\theta_{t-k}$ on $\theta_t$ is $A_{t-k,t} = \delta^{k} I$; (d) the filtering recurrences of Theorem 4.5 simplify to (i) $a_t (-k) = a_{t-1} (-k + 1) + \delta^{k} A_t e_t$, $R_t (-k) = R_{t-1} (-k + 1) - \delta^{2k} A_t Q_t A_t'$, (ii) $a_t (-k) = m_{t-k} + \delta[a_t (-k + 1) - a_{t-k+1}]$, $R_t (-k) = C_{t-k} + \delta^{2} [R_t (-k + 1) - R_{t-k+1}]$. :::: ::: {#exr-ch4-ex12} Generalise Theorem 4.1 to the multivariate DLM $\{F_t , G_t , V_t , W_t \}$. ::: ::: {#exr-ch4-ex13} Verify the filtering and smoothing results in Corollaries 4.3 and 4.4 relating to the DLM with an unknown constant observational variance $V$ . ::: ::: {#exr-ch4-ex14} Prove the reference prior results stated in part (1) of Theorem 4.12, that relate to the case of a known observational variance V . ::: ::: {#exr-ch4-ex15} ### Reference analysis updating for the first-order polynomial model Prove the reference prior results stated in part (2) of Theorem 4.13, relating to the case of an unknown observational variance V. ::: ::: {#exr-ch4-ex16} ### Reference analysis updating for the first-order polynomial model Prove the results stated in Corollary 4.8, providing the conversion from reference analysis updating to the usual recurrence equations when posteriors become proper. ::: ::: {#exr-ch4-ex17} ### Reference analysis updating for the first-order polynomial model Consider the first-order, polynomial model $\{1, 1, V, W\}$, with $n = 1$ and $\theta_t = \mu_t$, in the reference analysis updating of Section 4.10. (a) Assume that $V$ is known. Using the known variance results of Theorem 4.12 and Corollary 4.8, show that $(\mu_1 \mid \mathcal{D}_1) \sim \mathcal{N}[Y_1, V]$. (b) Assume that $V = 1/\phi$ is unknown, so that the results of Theorem 4.13 and Corollary 4.8 apply. Show that the posterior for $\mu_t$ and $V$ becomes proper and of standard form at $t = 2$, and identify the defining quantities $m_2$, $C_2$, $n_2$ and $S_2$. (c) In (b), show directly how the results simplify in the case $W = 0$. ::: ::: {#exr-ch4-ex18} ### Reference analysis updating for the 2-dimensional model Consider the 2-dimensional model $$ \left\{ \begin{pmatrix} 1 & 0 \end{pmatrix}, \begin{pmatrix} 1 & 1 \\[4pt] 0 & 1 \end{pmatrix}, V, 0 \right\} $$ assuming $V$ to be known. Apply Theorem 4.13 and Corollary 4.8 to deduce the posterior for $(\theta_2 \mid \mathcal{D}_2)$. ::: ::: {#exr-ch4-ex19} ### Discount regression DLM with unknown variance Consider the discount DLM $\{F_t , I, V, V W_t^*\}$ with unknown but constant variance $V$. The discount factor is $0 < \delta \le 1$, so that $W_t^* = C_{t-1}^* (1 - \delta)/\delta$, and $R_t^{*-1} = \delta C_{t-1}^{*-1}$. Initially, ignorance is formulated so the prior precision of $\theta_0$ is $C_0^{-1} = 0$, the prior degrees of freedom for $\phi = 1/V$ are $n_0 = 0$, $m_0 = 0$ and $S_0 = 0$. Use the notation of Theorem 4.3 so that starred variances are conditioned on $V = 1$. (a) Prove that $C_1^{*-1} = F_1 F_1'$. (b) Prove that $$ C_t^{*-1} = \sum_{v=0}^{t-1} \delta^{v}\, F_{t-v} F_{t-v}'. $$ (c) Suppose that $\operatorname{rank}\, C_t^{*-1} = \operatorname{rank}\, C_{t-1}^{*-1} = r_{t-1}$. Show that the relationship between $Y_t$ and the information $\mathcal{D}_{t-1}$ can be modelled by a DLM $\{\tilde{F}_t , I, V, V \tilde{W}_t \}$ with a parameter of dimension $r_{t-1}$ that, conditional upon $V$, has a proper distribution. Con- sequently, show that a conditional forecast exists such that $(Y_t \mid \mathcal{D}_{t-1}, V) \sim \mathcal{N}[f_t , V Q_t^*]$, and that given $Y_t$, the variance can be updated according to $n_t = n_{t-1} + 1$, $S_t = S_{t-1} + (e_t^{2}/Q_t^* - S_{t-1})/n_t$, with $n_1 = 1$ and $S_1 = e_1^{2}$. Show further that if $n_{t-1} > 0$, the unconditional forecast $(Y_t \mid \mathcal{D}_{t-1}) \sim \mathcal{T}_{n_{t-1}}[f_t , Q_t]$ exists. Suggest a method of obtaining $m_t$ for this reduced DLM. (d) Now suppose $\operatorname{rank}\, C_t^{*-1} = 1 + \operatorname{rank}\, C_{t-1}^{*-1}$. Show that no forecast of $(Y_t \mid \mathcal{D}_{t-1}, V)$ is possible. However, given $Y_t$, show that although $\{n_t , S_t \} = \{n_{t-1} , S_{t-1} \}$, the dimension of the design space for which forecasts can now be derived is increased by $1$. (e) Collinearity can be a real problem. For example, in the simplest regression discount DLM, a price variable used as a regressor may be held constant for quite a long period before being changed. Obviously, information is being gathered, so that useful conditional forecasts can be made based upon this price. However, forecasts conditional on other prices cannot be made until after a price change has been experienced. In general, at time $t - 1$, the forecast design space is spanned by $F_1 , \ldots , F_{t-1}$. Construct an algorithm to provide a reference analysis accommodating collinearity that at time $t - 1$, enables forecasts and monitoring for time $t$, whenever $F_t$ is contained in the current forecast design space and $n_{t-1} \ge 1$. (f) Generalise this approach beyond discount regression DLMs. Some relevant work is presented in Vasconcellos (1992). :::