Local linear trend model

Definition

The local linear trend block describes the following trend component:

\[l_{t+1} = l_t + n_t + \xi_t\] \[n_{t+1} = n_t + \mu_t\] \[\xi_t \sim N(0, \sigma^2\sigma^2_l)\] \[\mu_t \sim N(0, \sigma^2\sigma^2_n)\]

It conforms to the general SSF by setting the state vector

\[\alpha_t=\begin{pmatrix} l_t \\ n_t \end{pmatrix}\]

and the transition matrices

\[T_t = \begin{pmatrix} 1 && 1 \\ 0 && 1 \end{pmatrix}\] \[V_t = \begin{pmatrix} \sigma^2\sigma^2_l && 0 \\ 0 && \sigma^2\sigma^2_n \end{pmatrix}\]

Depending on the estimation algorithm used, it may be preferrable to slightly modify the model and set $\sigma^2=1$

jd3_ssf_locallineartrend

R users can use this function to define local linear trend model that belongs to the JD+ class JD3_SsfItem

output

Object of the JD+ class JD3_SsfItem:

usage

jd3_ssf_locallineartrend(name, levelVariance, slopevariance, fixedLevelVariance, fixedSlopeVariance )

Argument	Definition	Default	Remarks
name	string with the name of the component		ex. "trend", do not forget the commmas
levelVariance	double for the value of $\sigma^2_l$	1	variance of the level innovation is actually $\sigma^2\sigma^2_l$
slopeVariance	double for the value of $\sigma^2_n$	0.01	variance of the slope innovation is actually $\sigma^2\sigma^2_n$
fixedLevelVariance	logical that triggers estimation of $\sigma^2_l$ (FALSE) or fixes it (TRUE) to a pre-specified value set by the parameter levelVariance	FALSE
fixedSlopeVariance	logical that triggers estimation of $\sigma^2_n$ (FALSE) or fixes it (TRUE) to a pre-specified value set by the parameter slopeVariance	FALSE

Examples of use

jd3_ssf_locallineartrend("trend")
jd3_ssf_locallineartrend("trend", levelVariance=1, slopeVariance=0.001,fixedLevelVariance=FALSE, fixedSlopeVariance=FALSE) // default
jd3_ssf_locallineartrend("trend", levelVariance=1, slopeVariance=0.001,fixedLevelVariance=TRUE, fixedSlopeVariance=FALSE) // default

Application

In this example we design a model based on a random walk with drift specification.

Create Model:

The very first thing one needs to do is to create the model object, which will be called “yourModel”:
```
yourModel<-jd3_ssf_model()
```

Specify latent variables (transition equations):

Second, we add our random walk with drift and name it as “trend”. We fix the slope variance to zero in order to ensure that this component becomes a constant. In the absence of further arguments the variance of the level is not specified among the arguments and it is therefore the only parameter to estimate (by default, fixedLevelVariance=FALSE):
```
add(yourModel, jd3_ssf_locallineartrend("trend", slopeVariance=0, fixedSlopeVariance=TRUE) )
```

Measurement equation design:

Let’s define the following measurement equation (see Measurements)

\[eq1 : y_{t} = l_t + \epsilon_{1,t}\] \[eq2 : y_{t+h|t} = l_t + h c + \epsilon_{2,t}\]

The first equation represents an observed variable $y_{t}$ that measures (with an error $\epsilon_{1,t}$) the latent factor $l_t $, which is assumed to follow a random walk with drift. The second variable represents a h-steps ahead forecast for $y_{t}$, which is given by the h-steps ahead forecast of the trend $l_t + h c $. Note that $l_t$ and $n_{t}=c$ are the two elements of the state vector in the local linear trend model defined above. The subindex $t$ in $n_{t}$ in not necessary because we have set the slopeVariance to zero and therefore this term is constant over time (i.e. it represents the constant drift component).

Let’s assign the names “eq1” and “eq2” to our measurement equations, where the time series are called “y” ($ y_{t} $) and “yh” ($ y_{t+h|t} $) The I.I.D. measurement error component $var(\epsilon_{1,t})=\sigma^2 \sigma^2_{\epsilon_1}$ will be identified using the normalization assumption that $\sigma^2_{\epsilon_{1}}=1$. In turn, $var(\epsilon_{2,t})$ is left unrestricted.

eq1 <-jd3_ssf_equation("y", variance=1, fixed=TRUE)                            
add(eq1, "trend")                                         
eq2 <-jd3_ssf_equation("yh", variance=1, fixed=FALSE)                            
add(eq2, "trend", jd3_ssf_loadings(c(1,2),c(1,h)) )                                         