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I am trying to estimate the following state space model:

\begin{equation} y_t = y^{gap}_{t} + y^*_t \end{equation} \begin{equation} y^{gap}_{t} = \alpha_{1}y^{gap}_{t-1}+\alpha_{3}y^{gap}_{t-2} +\alpha_{2}/2(r_{t-1}-r^*_{t-1}) + \alpha_{2}/2(r_{t-2}-r^*_{t-2}) + \epsilon^{ygap}_t \end{equation} \begin{equation} y^*_t = y^*_{t-1}+ \mu_{t-1} + \epsilon^{y^*}_t \end{equation} \begin{equation} \mu_t = \mu_{t-1} + \epsilon^{\mu}_t \end{equation} \begin{equation} u_t = u^{gap}_{t} + u^*_t \end{equation} \begin{equation} u^{gap}_{t} = \gamma_10.4y^{gap}_{t}+\gamma_10.3y^{gap}_{t-1}+\gamma_10.2y^{gap}_{t-2}+\gamma_10.1y^{gap}_{t-3} +\epsilon^{u^{gap}}_t \end{equation} \begin{equation} u^*_t = u^*_{t-1} + \epsilon^{u^*}_t \end{equation} \begin{equation} \pi_{t} = \beta_{1}/3\pi_{t-1} + \beta_{1}/3\pi_{t-2} + \beta_{1}/3\pi_{t-3} + \beta_{2}u^{gap}_{t-1} + (1-\beta_{1})\pi^{e}_{t} +\epsilon^{\pi}_t \end{equation} \begin{equation} r^*_t = 4\mu_t+z_{t} \end{equation} \begin{equation} z_t = z_{t-1} +\epsilon^{z}_t \end{equation}

However, the results I am getting are completely wrong. For example, my output gap is trending like a non-stationary variable.

I have had a crack a casting this model into a form suitable for R's DLM package:

\begin{equation} Y_t = F\theta_t + V_t \end{equation} \begin{equation} \theta_t = G\theta_{t-1} + W_t \end{equation}

Where

\begin{equation} Y = \left[\begin{array}{c} y_{t} \\ u_{t} \\ r_{t} \\ \pi_{t} \end{array}\right] \end{equation}

\begin{equation} F = \left[\begin{array}{ccccccccccccccccc} 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ \end{array}\right] \end{equation}

\begin{equation} \theta_{t} = \left[\begin{array}{c} y^* \\y^{gap}_{t} \\y^{gap}_{t-1} \\y^{gap}_{t-2} \\ \mu_{t} \\z_{t}\\ r^*_{t} \\ r^*_{t-1}\\ r_{t} \\ r_{t-1} \\ u^*_{t}\\ u^{gap}_{t}\\ u^{gap}_{t-1} \\ \pi_{t} \\\pi_{t-1} \\ \pi_{t-3} \\ 1-\beta_1 \end{array} \right] \end{equation}

\begin{equation} G =\left[\begin{array}{ccccccccccccccccc} 1& 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & \alpha_1 & \alpha_2 & 0 & 0 & 0 & \alpha_3/2 & \alpha_3/2 & \alpha_3/2 & \alpha_3/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 &0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 4 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & \gamma_1(0.4\alpha_1+0.3) & \gamma_1(\alpha_20.4+0.2) & \gamma_10.1 & 0 & 0 & \gamma_10.4\alpha_3/2 & \gamma_10.4\alpha_3/2 & \gamma_10.4\alpha_3/2 & \gamma_10.4\alpha_3/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \beta_3 & \beta_1/3 & \beta_1/3 & \beta_1/3 & \pi^{e}_{t} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ \end{array}\right] \end{equation} \begin{equation} R =\left[\begin{array}{ccccccccccccccccc} \sigma^{y^*} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & \sigma^{y^{gap}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & \sigma^{\mu} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & \sigma^{z} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 4\sigma^{\mu} & \sigma^{z} & 4\sigma^{\mu}+\sigma^{z} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \sigma^{r_t} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \sigma^{u^{*}_t} & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & \gamma_1\sigma^{y^{gap}_t} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \sigma^{u^{gap}_t}+\gamma_1\sigma^{y^{gap}_t} & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \sigma^{\pi} & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{array}\right] \end{equation} \begin{equation} W=RR^T \end{equation}

As the above shows, I am using a little trick by treating r as a random walk - as advised by the author of the DLM package in response to a question: https://r.789695.n4.nabble.com/Setting-up-a-State-Space-Model-in-dlm-td3580664.html

My code is below, I have used the estimates from this paper for starting values of the coefficients: https://www.rba.gov.au/publications/bulletin/2017/sep/pdf/bu-0917-2-the-neutral-interest-rate.pdf

I apologise as this is not 100% reproducable as I am not sure how to post the data for the model - I am happy to share! Code:



NRDLM <- dlm(

  FF = matrix(0,4,17),

  V = diag(0.00001, 4),

  GG =diag(0,17),

  JGG = diag(1,17),

  W = diag(0,17),

  m0 = rep(0,17),

  C0 = diag(1000000,17),

  X = NRdata[,c("Inflation.e")]

)

# Matrix to parametrise VCV matrix W
R <- diag(0,17)

# Set all elements of JGG to zero (will change below)
NRDLM$JGG <- diag(0,17)

# build DLM
buildNRDLM <- function(p){

  FF(NRDLM)[1,1] <- 1
  FF(NRDLM)[1,2] <- 1

  FF(NRDLM)[2,11] <- 1
  FF(NRDLM)[2,12] <- 1

  FF(NRDLM)[3,9] <- 1

  FF(NRDLM)[4,14] <- 1

  GG(NRDLM)[1,1]  <- 1 
  GG(NRDLM)[1,5]  <- 1
  GG(NRDLM)[2,2]  <- p[1] 
  GG(NRDLM)[2,3]  <- p[2]
  GG(NRDLM)[2,7]  <- p[3]/2
  GG(NRDLM)[2,8]  <- p[3]/2
  GG(NRDLM)[2,9]  <- -p[3]/2
  GG(NRDLM)[2,10]  <- -p[3]/2
  GG(NRDLM)[3,2]  <- 1
  GG(NRDLM)[4,3]  <- 1


  GG(NRDLM)[5,5]  <- 1 
  GG(NRDLM)[6,6]  <- 1 
  GG(NRDLM)[7,5]  <- 4 
  GG(NRDLM)[7,6]  <- 1 
  GG(NRDLM)[8,7]  <- 1 

  GG(NRDLM)[9,9]  <- 1 
  GG(NRDLM)[10,9]  <- 1 

  GG(NRDLM)[11,11] <- 1

  GG(NRDLM)[12,2]  <- p[4]*(0.4*p[1]+0.3)        
  GG(NRDLM)[12,3]  <- p[4]*(0.4*p[2]+0.2)        
  GG(NRDLM)[12,4]  <- p[4]*0.1       
  GG(NRDLM)[12,7]  <- p[4]*0.4*p[3]/2       
  GG(NRDLM)[12,8]  <- p[4]*0.4*p[3]/2       
  GG(NRDLM)[12,9] <- p[4]*0.4*-p[3]/2            
  GG(NRDLM)[12,10] <- p[4]*0.4*-p[3]/2           
  GG(NRDLM)[13,12] <- 1           

  GG(NRDLM)[14,13] <- p[5]
  GG(NRDLM)[14,14] <- p[6]/3
  GG(NRDLM)[14,15] <- p[6]/3
  GG(NRDLM)[14,16] <- p[6]/3
  GG(NRDLM)[15,14] <- 1
  GG(NRDLM)[16,15] <- 1
  GG(NRDLM)[14,17] <- 1

  JGG(NRDLM)[14,17] <- 1


  # Variance covariance - RR'

  R[1,1] <- p[7]
  R[2,2] <- p[8]
  R[5,5] <- p[9]
  R[6,6] <- p[10]
  R[6,7] <- p[10]
  R[7,5] <- 4*p[9]
  R[9,9] <- p[11]
  R[11,11] <- p[12]
  R[12,12] <- p[13]
  R[12,2] <- p[4]*p[8]
  R[14,14] <- p[14]

  W(NRDLM) <- R%*%t(R)


  m0(NRDLM) <- c(NRdata$log.output[1],0,0,0,mean(diff(NRdata$log.output[1:4])),0,NRdata$real.r[2],NRdata$real.r[1],NRdata$real.r[2],NRdata$real.r[1],NRdata$unr[1],0,0,NRdata$Inflation[3],NRdata$Inflation[2],NRdata$Inflation[1],0)


  return(NRDLM)

}

theta <- c(1.53,-0.54, -0.05, 0.62, -0.32, 0.39, 0.38, 0.54, 0.05, 0.22, 0 , 0.15, 0.07, 0.79 ) # estimates from paper

# Estimate model

NRDLM.est <-  dlmMLE(y = cbind(NRdata$log.output,NRdata$unr,NRdata$real.r,NRdata$Inflation), parm = theta, build = buildNRDLM, lower =c(rep(-Inf,6),rep(exp(-8),7)), upper= c(rep(Inf,6),rep(exp(12),11)),
                     control = list(trace = 1, REPORT = 5, maxit = 1000), hessian = TRUE, debug = F)




#--------------------------------------------------------------------------------------------------------------------------
# filtered and smoothed estimates
#--------------------------------------------------------------------------------------------------------------------------

NRDLMbuilt <- buildNRDLM(NRDLM.est$par)

filtered <- dlmFilter(y =cbind(NRdata$log.output,NRdata$unr,NRdata$real.r,NRdata$Inflation), mod = NRDLMbuilt)

smoothed <- dlmSmooth(y = cbind(NRdata$log.output,NRdata$unr,NRdata$real.r,NRdata$Inflation), mod = NRDLMbuilt)

Anything obviously wrong here?

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  • $\begingroup$ I would think the $W$ you are providing is rather a factor of the true W, for it does not appear to be symmetric. $\endgroup$
    – F. Tusell
    Commented Apr 11, 2020 at 10:23
  • $\begingroup$ @F.Tusell you're correct - I have updated my question to reflect this. I am using a matrix $R$ which i then use to calculate $W$ as $RR^T$. I'm actually struggling a little with how I should correctly parameterize W, I think I'll need ygaps shock to drive ugaps transition, as well as ugaps own shock (since the contemporaneous correlation between ugap and ygap) I also know that I need to drive r* with the shocks driving mu and z. But I don't think I've done this quite right and I'm not sure how to proceed. – $\endgroup$
    – Adam_123
    Commented Apr 11, 2020 at 13:32
  • $\begingroup$ Would not like to sound ominous, but my own experience is that with state vectors this big and having to estimate lots of parameters, life is hard. Even assuming your code is correct, many things can go wrong. You should try with a variety of starting points, with different algorithms (Nelder-Mead can be slow, but on certain problems with poor differentiability has given more stable results to me than BFGS). Also, although I do not know your problem, I suspect there must be some constraint on $\beta_1$ which should be accounted for. $\endgroup$
    – F. Tusell
    Commented Apr 11, 2020 at 15:55
  • $\begingroup$ @F.Tusell thanks for taking a look, I appreciate it. I agree, life on this on has been hard. But I think this is achievable, there are many papers that do variations of this type of model (it is a variation of the rather famous Labauch and William's model) there are constraints put on the estimates of the variance in their example too, due to the limitations of MLE. The paper I link to below uses a Gibbs sampling approach, which was my next step. But I thought MLE would give me something close to the correct answer. Unfortunately that's not the case. I'll keep persevering! $\endgroup$
    – Adam_123
    Commented Apr 11, 2020 at 23:13
  • $\begingroup$ @F.Tusell estimation techniques aside, does the SS model it self look correct? Going from the equation written out in their normal form to casting into a SS model. A lot of the other papers I have seen have used a different version of SS model - one that explicitly allows for exogenous variables, I thought it would be a good challenge to adapt the model to one suitable for DLM. $\endgroup$
    – Adam_123
    Commented Apr 11, 2020 at 23:18

1 Answer 1

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I've reworked the model and think I have landed somewhere sensible. My results are in line with other studies, which is a good sign. To get this model to work, I had to exogenously estimate some the variances - denoted $\lambda$ in the $W$ matrix.

The matrices are as follows: \begin{equation} Y_t = \left[\begin{array}{c} y_{t} \\ u_{t} \\ \pi_{t} \\ \pi^{e}_{t}\\ r_{t} \\ \end{array}\right] \end{equation} \begin{equation} F = \left[\begin{array}{ccccccccccccccccc} 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{array}\right] \end{equation} \begin{equation} \theta_t =\left[\begin{array}{c} y* \\y^{gap}_{t} \\y^{gap}_{t-1} \\y^{gap}_{t-2} \\ \mu_{t} \\\mu_{t-1}\\ r_{t} \\ r_{t-1}\\ z_{t} \\ z_{t-1} \\ u^*_{t}\\ u^{gap}_{t}\\ u^{gap}_{t-1} \\ \pi_{t} \\\pi_{t-1} \\ \pi_{t-2} \\ \pi^{e}_t \end{array} \right] \end{equation}

\begin{equation} G =\left[\begin{array}{ccccccccccccccccc} 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & \alpha_1 & \alpha_2 & 0& 0 & -c\alpha_3/2 & -c\alpha_3/2 & -\alpha_3/2 & -\alpha_3/2 & \alpha_3/2 & \alpha_3/2 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & U_1 & U_2 & U_3 & 0 & U_4 & U_5 & U_6 & U_7 & U_8 & U_9 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \beta_2 & \beta_1/3 & \beta_1/3 & \beta_1/3 & (1-\beta_1) \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right]\end{equation}

Where \begin{equation} U_1 = \gamma_1(0.4\alpha_1+0.3), U_2 =\gamma_1(\alpha_20.4+0.2), U_3 = \gamma_10.1, U_4 = -\gamma_10.4c\alpha_3/2 \\ U_5 =-\gamma_10.4c\alpha_3/2, U_6 = -\gamma_10.4\alpha_3/2, U_7 = -\gamma_10.4\alpha_3/2, U_8 = \gamma_10.4\alpha_3/2 , U_9 = \gamma_10.4\alpha_3/2 \end{equation}

\begin{equation} W_t =\left[\begin{array}{ccccccccccccccccc} (1+\lambda_g^2)\sigma_{y^*}^2 & 0 & 0 & 0 & (\lambda_g\sigma_{y^*})^2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & \sigma_{y^{gap}}^2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & (\gamma_1\sigma_{y^{gap}})^2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ (\lambda_g\sigma_{y^*})^2 & 0 & 0 & 0 & (\lambda_g\sigma_{y^*})^2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 &0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & \sigma_{r}^2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & (\frac{\lambda_z\sigma_{y^{gap}}}{a_3})^2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \sigma_{u^{*}_t}^2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & (\gamma_1\sigma_{y^{gap}})^2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \sigma_{u^{gap}}^2+(\gamma_1\sigma_{y_{gap}})^2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \sigma_{\pi}^2+((1-\beta_1)\sigma_{\pi^{e}})^2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \sigma_{\pi^{e}}^2 \end{array}\right] \end{equation}

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