multivariate state space model dlm okuns law

Question

I'm trying to estimate an Okun's law equation with a dlm using the dlm package in R. I can estimate the non-time varying model using nls as follows:

const_coef <- nls(formula = dur~ b1*dur_lag1 + b2*(d2lgdp-b0) + b3*d2lrulc_lag2 ,
start = list(b0 =0.1, b1=0.1, b2=0.1, b3=0.1),
data = mod_data) 

the dlm model I want to be able to estimate allows for b1 and b0 in the above to follow random walks. I can do this in Eviews by declaring the measurement equation and appending the states (below is some code provided by the authors of the original paper which I can replicate:

'==========================
' SPECIFY THE KALMAN FILTER
'==========================

'Priors on state variables
vector(2) mprior
    mprior(1) = 4           'Prior on starting value for trend GDP growth (annual average GDP growth over 1950s)
    mprior(2) = 0           'Prior on starting value for lagged dependent variable
sym(2) vprior
    vprior(1,1) = 5         'Prior on variance of trend GDP growth   (variance of annual GDP growth over 1950s)
    vprior(2,2) = 1         'Prior on variance of lagged dependent variable

'Specify coefficient vector
coef(8) ckf

'Declare state space
sspace ss1
ss1.append dur = lag*dur(-1) + ckf(2)*(d2lgdp-trend)+ckf(3)*D2LRULC(-2)+[var=exp(ckf(4))]   'Measurement equation
ss1.append @state trend = 1*trend(-1) + [var = exp(ckf(5))]                                                 'State equation for trend GDP growth (random walk)
ss1.append @state lag = 1*lag(-1) + [var = exp(ckf(6))]                                                     'State equation for lagged dependent variable (random walk)

'Apply priors to state space
ss1.append @mprior mprior
ss1.append @vprior vprior

'Set parameter starting values
param ckf(2) -0.0495 ckf(3) 0.01942 ckf(4) -2.8913 ckf(5) -4.1757 ckf(6) -6.2466        'starting values for parameters

'=====================
' ESTIMATE THE MODEL
'=====================

'Estimate state space
smpl %estsd %ested          'Estimation sample
ss1.ml(m=500,showopts)      'Estimate Kalman filter by maximum likelihood
freeze(mytab) ss1.stats

I'm really not sure how to do this with the dlm package. I've tried the following:

buildSS <- function(v){


  dV <- exp(v[1])               # Variance of the measurment equation (ckf4)
  dW <- c(exp(v[2]),            # variance of the lagged dep  (ckf6)
          0,                    # variance of the coef on d2lgdp ckf(2) set to 0
          0,                    # variance of the coef on d2lrulc ckf(3) set to 0
          exp(v[3])             # variance of the random walk intercept (ckf5)
           )

  beta.vec <- c(1,v[4],v[5],1)           # Params ckf(2)  ckf3(3)


  okuns <- dlmModReg(mod_data.tvp[,-1], addInt = TRUE, dV =dV, dW = dW, m0 = beta.vec)




}

#'Set parameter starting values

ckf4Guess <- -2.8913
ckf2guess <- -0.0495
ckf3guess <- 0.01942
ckf5guess <- -4.1757
ckf6guess <- -6.2466



params <- c(ckf4Guess,
            ckf5guess,
            ckf6guess,
            ckf2guess,
            ckf3guess)

tvp_mod.mle <- dlmMLE(mod_data.tvp[,"dur"] , parm = params, build = buildSS)

tvp_mod <- buildSS(tvp_mod.mle$par)

tvp_filter <-  dlmFilter(mod_data$dur,tvp_mod)

The above code runs, but the outputs are not correct. I am not specifying the the states properly. Does anyone have any experience in building dlms with mutlvirate regression in R?

Answer 1

I think I have gotten to a solution - I've managed to recreate the estimates in the paper which estimates this model using Eviews (also checked this using Eviews).

    #--------------------------------------------------------------------------------------------------------------------------
# tvp model full model - dur = alpha*dur(-1)+ beta(dgdp-potential) + gamma*wages
#--------------------------------------------------------------------------------------------------------------------------

# Construct DLM

OkunsDLMfm <- dlm(


  FF = matrix(c(1,1,1,1),ncol = 4, byrow = TRUE),

  V = matrix(1),

  GG = matrix(c(1,0,0,0,
                0,1,0,0,
                0,0,1,0,
                0,0,0,1), ncol = 4, byrow = TRUE),

  W =  matrix(c(1,0,0,0,
                0,1,0,0,
                0,0,1,0,
                0,0,0,1), ncol = 4, byrow = TRUE),

  JFF = matrix(c(1,2,3,0),ncol = 4, byrow = TRUE),

  X = cbind(mod_data$dur_lag1,mod_data$d2lgdp, mod_data$d2lrulc_lag2), # lagged dep var, dgdp, wages.

  m0 = c(0,0,0,0),

  C0 = matrix(c(1e+07,0,0,0,
                0,1e+07,0,0,
                0,0,1e+07,0,
                0,0,0,1e+07), ncol = 4, byrow = TRUE)

)


buildOkunsFM <- function(p){

  V(OkunsDLMfm)  <- exp(p[2])

  GG(OkunsDLMfm)[1,1]  <- 1

  GG(OkunsDLMfm)[2,2]  <- 1

  GG(OkunsDLMfm)[3,3]  <- 1 

  GG(OkunsDLMfm)[4,4]  <- 1

  W(OkunsDLMfm)[1,1] <- exp(p[3])

  W(OkunsDLMfm)[2,2] <- 0

  W(OkunsDLMfm)[3,3] <- 0

  W(OkunsDLMfm)[4,4] <- exp(p[4])

  m0(OkunsDLMfm) <- c(0,0,0,p[1]*4)

  C0(OkunsDLMfm)[1,1] <- 1

  C0(OkunsDLMfm)[4,4] <- 5


  return(OkunsDLMfm)

}



okuns.estfm <-  dlmMLE(y = mod_data$dur, parm = c(-0.049,-1.4,-6,-5), build = buildOkunsFM)


OkunsDLM1fm <- buildOkunsFM(okuns.estfm$par)

The time varying level, the estimate of potential output, is derived by dividing the 4 element of the state vector by the second * by negative 1.

Not sure if this is best way to specify the DLM, but the results from the model are very close to what is reported (within 0.01) of the results from using Eviews. That being said, very open to any other specifications.