Local level (random walk) model

Definition

The local level block describes a random walk component. When the innovation variance is set to 0, it also describes a constant term (known when the initialization is specified, estimated otherwise)

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

It conforms to the general SSF by setting $ T_t = 1 $, $ V_t = \sigma^2\sigma^2_l $.

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

jd3_ssf_locallevel

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

output

Object of the JD+ class JD3_SsfItem:

usage

jd3_ssf_locallevel(name, variance, fixed, initial)

Argument	Definition	Default	Remarks
name	string with the name of the component		ex. "trend", do not forget the commmas
variance	double for the value of $\sigma^2_l$	1	variance of the innovation is actually $\sigma^2\sigma^2_l$
fixed	logical that triggers estimation of $\sigma^2_l$ (FALSE) or fixes it (TRUE) to a pre-specified value set by the parameter variance	FALSE
initial	double setting the value of $l_0$	NaN	The default NaN triggers the [diffuse initialization](../algorithms/dk.html)

Examples of use

jd3_ssf_locallevel("trend")
jd3_ssf_locallevel("trend", variance=1, fixed=FALSE, initial=NaN) // default
jd3_ssf_locallevel("trend", variance=1, fixed=TRUE)               // fix variance to 1
jd3_ssf_locallevel("trend", variance=1, fixed=TRUE, initial=0)    // fix variance to 1 and initial state to 0

Application 1

In this example we design a model based on a local level specification with default settings (i.e. random walk)

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 and name it as “trend”. In the absence of further arguments, this function triggers the default specification, which is a random walk without drift:
```
add(yourModel, jd3_ssf_locallevel("trend"))
```

Measurement equation design:

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

\[eq1 : y_{t} = l_t + \epsilon_{t}\]

where $\epsilon_{t}$ is I.I.D. with $var(\epsilon_{t}) =\sigma^2 \sigma^2_{\epsilon}$

Let’s assign the name “eq1” to our measurement equation, where the time series called “yourData” ($y_{t}$) is assumed to depend on the factor “trend” ($l_t$). By default, the trend enters with coefficient equal to 1. The I.I.D. measurement error component $\epsilon$ has a variance equal to zero by default (i.e. $var(\epsilon_{t})=\sigma^2 \sigma^2_{\epsilon}$ where $\sigma^2_{\epsilon} =0$ by default):
```
eq1 <-jd3_ssf_equation("yourData")                            
add(eq1, "trend")                                         
```
Since $var(\epsilon_{t}) =\sigma^2 \sigma^2_{\epsilon}$ , the measurement error component is exactly identified by fixing $\sigma^2_{\epsilon}=1$. However, we have said that the default specification fixes $\sigma^2_{\epsilon}=0$ . In order to impose the restriction $\sigma^2_{\epsilon}=1$, we use the following notation:
```
eq1 <-jd3_ssf_equation("yourData",variance=1, fixed = TRUE) )                            
add(eq1, "trend")                                         
add(eq1, "constant")                                         
```

By default, the coefficient associated to the factor (trend) is equal to one. In order to relax this assumption and estimate a different model: $eq1 : y_{t} =\alpha \tau_{t} + \epsilon_{t}$, we use the following arguments:

eq1 <-jd3_ssf_equation("yourData",variance=1, fixed = TRUE) )                            
add(eq1, "trend", 0.5, FALSE)                                         
add(eq1, "constant")                                         

where $\alpha$ is a free parameter that is set to $\alpha =0.5$ only for the purposes of initialization.

Application 2 (multivariate)

In this example we design a model with two observables and two factors. The first one is a local level specification with default settings (i.e. random walk) and the second factor is a new local level that degenerates to a constant term.

Thus, we have two different measurements $y_{1,t}, y_{2,t}$ for the level $l_t$. The two measurement equations follow:

$eq1 : y_{1,t} =c_{1} + \alpha_{1} l_{t} + \epsilon_{1,t}$,

$eq2 : y_{2,t} = \alpha_{2} l_{t} + \epsilon_{2,t}$,

This multivariate model would be specified as follows in R:

// create model

newModel<-jd3_ssf_model()

// specify latent variables

add(newModel, jd3_ssf_locallevel("tau"))
add(newModel, jd3_ssf_locallevel("c1", variance = 0, fixed = TRUE)) 

// two measurement equations

eq1 <-jd3_ssf_equation("y1",variance=1, fixed = TRUE) )                            
add(eq1, "tau", 0.5, FALSE)                                         
add(eq1, "c1")                                         

eq2 <-jd3_ssf_equation("y2",variance=1, fixed = FALSE) // by default it would be (variance=0, fixed = TRUE)                            
add(eq1, "tau", 0.5, FALSE)                                         

In this example, the second equation helps to identify $c_{1}$. The reason is that the initial value of $l_{t}$ and $c_{1}$ in the first equation cannot be separately identified.