 # State Space Model(s) and nowcasting

Hi everyone,
I am trying to do nowcasting by replicating the khakieconomics code. However, I get poor performance (at least I am not satisfied) in retrieving the model parameters (low ess_bulk, high \hat{R}).
Moreover, since these are the first steps for a larger project (multivariate response, gap-filling, and so on…) I would ask your opinion if I should use Gaussian Dynamic Linear Model. If it’s the case, I am struggling to understand the notation used in the manual, and maybe someone could point me to some stan code to start with because I can’t acces directly Durbin&Koopman’s book.
However, back to the original point, I reproduced the original code to simulate the data:

remove(list = ls())
library(dplyr)

set.seed(2)

N = 300
z1 = rnorm(n = N, mean = 0, sd = 3)
z2 = rnorm(n = N, mean = 0, sd = 3)
x = rep(NA_real_, N)
x = 1
for(i in 2:N) {
x[i] = x[i-1] + 0.4 * z1[i] + -0.3 * z2[i] + rnorm(n = 1, mean = 0, sd = 0.2)
}

time_step = seq(from = 0, length.out = N, by = 2)

freq = 28
y = x + rnorm(n = N, mean = 0, sd= 1)

y[!(1:N %% freq == 0)] = NA_real_

data = data.frame(TIME = time_step, Z1 = z1, Z2 = z2, X = x, Y = y)

y = y[!is.na(y)]

input = list(N1 = length(y), N2 = N, freq = freq, y = y, z1 = z1, z2 = z2)

library(cmdstanr)
library(posterior)
model = cmdstan_model("nowcasting.stan")
fit = model\$sample(data = input, seed = 123, chains = 4, parallel_chains = 4, refresh = 100, max_treedepth = 10)


and revamped the Stan model syntax

data {
int N1; // length of low frequency series
int N2; // length of high frequency series
int freq; // every freq-th observation of the high frequency series we get an observation of the low frequency one
vector[N1] y;
vector[N2] z1;
vector[N2] z2;
}

parameters{
vector gamma;
real<lower = 0> sigma_eta;
real<lower = 0> sigma_epsilon;
vector[N2] x;
}

model {
int count = 0;
x ~ normal(1,1);

for (i in 2:N2) {
x[i] ~ normal(x[i-1] +z1[i] * gamma + z2[i] * gamma, sigma_eta);
if(i%freq==0){
count += 1;
y[count] ~ normal(x[i], sigma_epsilon);
}

}
gamma ~ std_normal();
sigma_eta ~ cauchy(0, 1);
sigma_epsilon ~ cauchy(0, 1);

}


with these results:

Warning: 11 of 4000 (0.0%) transitions ended with a divergence.
This may indicate insufficient exploration of the posterior distribution.
Possible remedies include:
* Increasing adapt_delta closer to 1 (default is 0.8)
* Reparameterizing the model (e.g. using a non-centered parameterization)
* Using informative or weakly informative prior distributions

356 of 4000 (9.0%) transitions hit the maximum treedepth limit of 10 or 2^10-1 leapfrog steps.
Trajectories that are prematurely terminated due to this limit will result in slow exploration.
Increasing the max_treedepth limit can avoid this at the expense of more computation.
If increasing max_treedepth does not remove warnings, try to reparameterize the model.
> fit
variable   mean median     sd    mad    q5    q95 rhat ess_bulk ess_tail
lp__          219.80 205.31 138.65 111.75 45.27 499.16 1.19       16       NA
gamma        0.46   0.46   0.06   0.06  0.36   0.56 1.00     1460     1248
gamma       -0.30  -0.30   0.03   0.03 -0.35  -0.26 1.02     1555     1080
sigma_eta       0.32   0.31   0.12   0.12  0.11   0.52 1.18       16       NA
sigma_epsilon   0.85   0.78   0.56   0.57  0.10   1.83 1.03      103      485
x            0.01   0.03   0.92   0.92 -1.52   1.48 1.00     2953     3410
x            0.55   0.58   0.96   0.97 -1.05   2.07 1.00     2537     3248
x            1.95   1.96   0.97   0.96  0.34   3.48 1.00     2311     2732
x            2.09   2.10   1.05   1.05  0.36   3.71 1.01     2304     2822
x            1.31   1.31   1.05   1.03 -0.45   2.94 1.01     2758     2806

# showing 10 of 305 rows (change via 'max_rows' argument or 'cmdstanr_max_rows' option)
Warning message:
NAs introduced by coercion


As ever, any help is appreciated!

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Hi. A couple of things. The cauchy priors might have to heavy of tails. Can you try normal? And do you have a Kalman filter (or something similar) in place? The non-filter (naive) version of the state-space models can have some troubles.

There is a nice thread on this here: State space / kalman filtering advice for inverse model - #12 by lukas-rokka

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@Ara_Winter I have only translated the original code, so any advice for further experiments are welcomed and I definitely will try any path. So thanks! Moreover, I will dig into @lukas-rokka work

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I think the ctsem package implements a bunch of state-space model variants and might be a good inspiration…

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