SMC Procedure
INITSTATE Statement
INITSTATE state-variable ~ distribution </ option> ;
The INITSTATE statement specifies the initialization distribution of the state variable given the parameters; that is, 
. For more information about the initialization distribution such as the definitions of 
, p, and s, see the section Sequential Monte Carlo Methods and State Space Models. The INITSTATE statement is ignored if you specify the SUBMITMODEL statement. The INITSTATE statement has the following requirements if you omit the SUBMITMODEL statement:
The INITSTATE statement is required. Each state variable must have a corresponding INITSTATE statement.
Multiple INITSTATE statements are permitted. When you specify multiple INITSTATE statements, be careful about the order if there is any dependency between state variables.
The INITSTATE statement must have the following components in the order shown:
variable-name: A non-lagged or lagged format of a single state variable that you specify in the STATEVAR statement.
Tilde: A tilde (~) symbol between the state variable and the distribution.
-
Distribution: A distribution that is specified in the form distribution-name(distribution-arguments). The available distribution-name and distribution-arguments in the INITSTATE statements are displayed in Table 3. For the description of each distribution, see the section Standard Distributions.
Table 3: Distribution Names and Arguments
Distribution Name Distribution Arguments Definitions BETA a, b Beta distribution with arguments shape parameter a and shape parameter b CAUCHY l, s Cauchy distribution with arguments location l and scale s DEGENERATE c Degenerate distribution with argument c GAMMA a, b Gamma distribution with arguments shape a and scale b LOGN m, s Log-normal distribution with arguments mean m and standard deviation s NORMAL | N m, s Normal distribution with arguments mean m and standard deviation s POISSON 
Poisson distribution with argument mean
T m, s, 
Student’s —t distribution with arguments mean m, standard deviation s, and degrees of freedom
UNIFORM l, u Uniform distribution with arguments min l and scale u
The distribution-arguments are specified by expressions that are enclosed in parentheses and separated by commas. Such expressions must follow any SAS programming statements. To avoid the possible computation errors, you should make some adjustments to the expressions. For more information about the adjustments, see the section Distribution Argument Restriction. The number of expressions is equal to the number of arguments of the given distribution. For the INISTATE statement, the variables that can appear in the expressions must be selected from the following list:
the model parameters
the lagged dependent variables
the current and lagged independent variables
the current and lagged state variables that you specify in the preceding INITSTATE statements
the actual time index,
_time
For more information about the actual time index, see the section Time Index. For more information about the parameters, state variables, and dependent/independent variables, see the sections PARAMETERS Statement, STATEVAR Statement, and VAR Statement, respectively.
You can specify the following option after a forward slash (/):
-
PROPOSAL=proposal-dist
IMPORTANCE=proposal-dist -
specifies the proposal (importance) initialization distribution given the parameters, the dependent and independent variables at the first time step; that is,
. For more information about the proposal initialization equation in the state space model, see the section Sequential Monte Carlo Methods and State Space Models. If you omit this option, then the corresponding state variable uses the original initialization distribution as its proposal initialization distribution. In the SIR or APF algorithm, each state variable is sampled from its proposal initialization distributions at the first time step. Subsequently, the joint density of both the original and proposal initialization distributions of all the state variables is evaluated within the importance weight calculation. In the BF algorithm, the proposal initialization distribution of each state variable is always same as its original initialization distribution. For more information about the proposal initialization distribution, see the sections Sequential Importance Resampling Algorithm and Auxiliary Particle Filter. The format of the proposal-dist is proposal-dist-name(proposal-dist-arguments). The available proposal-dist-name and proposal-dist-arguments in the INITSTATE statements are also displayed in Table 3. The expressions for proposal-dist-arguments are enclosed in parentheses and separated by commas. Such expressions must follow any SAS programming statements. To avoid the possible computation errors, you should make some adjustments to the expressions. For more information about the adjustments, see the section Distribution Argument Restriction. The number of expressions is equal to the number of arguments of the given distribution type. The variables that are used in the expressions for proposal-dist-arguments must be selected from the following list:
the model parameters
the current and lagged dependent variables
the current and lagged independent variables
the current and lagged state variables that have been specified in the preceding INITSTATE statements
the actual time index,
_time
Moreover, the INITSTATE statements must satisfy the following conditions:
The maximum lag order of all the dependent variables in the distribution-arguments and the PROPOSAL= option of all INITSTATE statements must be less than or equal to p, where p is defined in the section Method 1: Maximum Lag Order of Dependent Variables, p.
The maximum lag order of all the independent variables in the distribution-arguments and the PROPOSAL= option of all INITSTATE statements must be less than or equal to s, where s is defined in the section Method 1: Maximum Lag Order of Independent Variables, s.
The variables that appear as the variable-name in all INITSTATE statements must not have duplicates; that is, each state variable or lagged state variable must appear only once as the variable-name in all INITSTATE statements.
Each state variable must appear once as the variable-name in one INITSTATE statement.
The maximum lag order of all the state variables in the variable-name of all the INITSTATE statements must be less than or equal to r, where r is defined in the section Method 1: Maximum Lag Order of State Variables, r
Suppose there are
state variables. For 
, if 
, where 
and 
are defined in the section Method 1: Maximum Lag Order of State Variables, r, then statev.L1, 
, statev.L
must appear once as the variable-name in one INITSTATE statement.
For example, suppose the initialization equations are
where 
. Then the following statements define these initialization equations:
initstate x1 ~ normal(0, gamma);
initstate x2 ~ normal(phi*x1, 1);
where phi and gamma are parameters and x1 and x2 are current state variables.
If 
has the proposal initialization equation
then the corresponding initialization distributions can be defined by
initstate x1 ~ normal(0, gamma) / proposal = normal(y, gamma);
initstate x2 ~ normal(phi*x1, 1);
where the state variable x1 has a proposal initialization distribution that incorporates the current dependent variable y, and the state variable x2 uses the original initialization as its proposal initialization distribution.