Random walk with drift in dynamic linear model

Question

Suppose I have a dynamic linear model as defined in the dlm-package for R, see Petris 2009.

$y_t = F_t θ_t + ν_t, ν_t$~$N(0,V_t)$

$θ_t = G_t θ_{t-1}+ω_t,ω_t$~$N(0,W_t)$.

A random walk without drift would be specified as $F_t=G_t=1$, resulting in

$y_t = θ_t + ν_t, ν_t$~$N(0,V_t)$

$θ_t = θ_{t-1}+ω_t,ω_t$~$N(0,W_t)$.

A “random walk with drift DLM” I would think of as

$y_t = θ_t + ν_t, ν_t$~$N(0,V_t)$

$θ_t = θ_{t-1}+d +ω_t,ω_t$~$N(0,W_t)$.

Can that be represented in the DLM framework defined as above? If so, how?

Possibly related: Random walk with drift are differences white noise? , as perhaps a such-specifiable DLM may be obtained by differencing the data.

Answer 1

Yes, it is possible with two states. Use dlmModPoly() to form a dlm from a second order polinomial. This initializes a model with the following parametrization:

$ y_t = \begin{pmatrix} 1 & 0 \end{pmatrix} \space \theta_t + \nu_t, \nu_t \sim N(0,V_t) \\ \theta_t = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \theta_{t−1} + ω_t, ω_t \sim N(0, W_t)$

where $\theta_t$ is a $2×1$ vector and $W_t$ is a $2×2$ covariance matrix.

Then estimate the time-invariant $V_t = x_1 $, $W_t=\begin{pmatrix} x_2 & 0 \\ 0 & 0 \end{pmatrix}$ (as well as the starting values $m_0$ for both states $\theta_{1,2})$.

This works as follows. At the observation level, the first state $\theta_{1}$ plus a normal observation noise is returned.

The first state equals its previous value plus the second state plus the normal random walk step.

As the second state is time-invariant (fixed variance of zero and autoregressive coefficient of 1), its value is entirely determined by the initial value $m_0$, which dlm solves using maximum likelihood estimation. This second state serves as the fix trend $d$.

Here is the R-code:

dlmModPoly(order = 2, dV = exp(x[1]), dW = c(exp(x[2]),0))