How to estimate NAIRU in a state space model of the econometrics toolbox

I would like to estimate the a time-varying nairu from the Philipps Curve: pi_t = pi*_t + beta*(u_t - u*_t) + e_t

pi_t: inflation

pi*_t: trend inflation

u_t: observed unemployment rate

u*_t: unobserved nairu

---------------------------------------------------

state equations are:

pi*_t = pi*_t-1 + v_1t

u*_t = u*_t-1 + v_2t

-------------------------------------------

 N = 184 %# observations
 m = 2 %# state equations  
 A = zeros(m);

    A(1,1) = 1;
    A(2,2) = NaN;

% Define the state-disturbance-loading matrix.

B = zeros(m);
  B(1,1) = 0.001;
  B(2,2) = 0.001;

% Define the measurement-sensitivity matrix.

   C = zeros(N,m);
    C(1,1) = 1;
    C(1,2) = 1;

% Define the observation-innovation matrix.

 D = zeros(N);
    D(1,1) = 0.025;

 params0 = -.3;

 StateType = [2;2];
 Mdl = ssm(A,B,C,D,'StateType',StateType);

Beta0   = [-.3];
 [EstMdl1,estParams,EstParamCov,logL,Output] ...
        = estimate(Mdl,yt,params0,'Predictors',Z,'Beta0',Beta0,'Display',{'params','diagnostics','full'})

    filteredX = filter(EstMdl1,yt,'Predictors',Z,'Beta',estParams(end));

Z = observed unemployment series ---------------------------------------------------------------------------------------

Above the code I have created and it does not give me any reasonable results.

Does anyone have any suggestions?

Answer 1

[Hang Qian]

It seems that the constructed SSM is not exactly the same as the one described in the equations. Usually C and D is a low-dimension matrix (say, C appears to be a 1-by-2 vector in this case), it is unnecessary to stack N observations in a giant matrix.

To check whether A,B,C,D construct a desired state-space model, consider disp(Mdl) and the model will be displayed on the screen equation by equation.

Also, I would not let the software guess both initial states, which are diffuse. I would incorporate priors on where the two random-walk states should start in that state-space model.

Regards,

- Hang Qian