# Dynamic panel data models with Stan?

**URL:** <https://discourse.mc-stan.org/t/dynamic-panel-data-models-with-stan/5136>\
**Category:** Modeling\
**Created:** [August 9, 2018, 2:09pm UTC](https://discourse.mc-stan.org/t/dynamic-panel-data-models-with-stan/5136 "2018-08-09T14:09:06Z")\
**Posts on this page:** 20\
**Page:** 1

<div class="post-metadata">

**Author:** ![James\_Savage](https://yyz2.discourse-cdn.com/flex030/user_avatar/discourse.mc-stan.org/james_savage/32/26_2.png) [@James\_Savage](https://discourse.mc-stan.org/u/James_Savage)\
**Post date:** [August 9, 2018, 2:09pm UTC](https://discourse.mc-stan.org/t/dynamic-panel-data-models-with-stan/5136/1 "2018-08-09T14:09:06Z")

</div>

It’s well known that in panel data models (that is, models with observations across individuals and time), if you have individual intercepts/slopes, then including a lagged dependent variable on the right hand side is a big no-no (potentially enormous biases).

To see this, imagine you have

y\_{it} = \alpha + u\_{i} + \delta y\_{it-1} + X\_{it}\beta + \epsilon\_{it}

with u\_{i} \sim \text{Normal}(0, \sigma\_{u}) being the random effects. By Gauss-Markov, these must be uncorrelated to X\_{it} and y\_{it-1} in our statistical model. But u\_{i} applies to all observations of y\_{i}, and so by construction E[u\_{i} y\_{it}]\neq 0. This typically inflates our estimate of \delta and shrinks our estimate of \beta.

Non-Bayesians use the Arellano-Bond estimator, which is a GMM approach to solving this problem. The idea is to express it in first-differences (getting rid of the random effect) in which case you’re only dealing with the correlation of the change in the error and change in lagged dependent variable. Then use GMM, instrumenting the differenced lagged dependent variable with its own lags.

Does anyone here have experience fitting these models in Stan? Happy to just throw Bayesian IV at it, but the Frequentist counterpart to this is less efficient than Arellano Bond. Ideas?

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<div class="post-metadata">

**Author:** ![anon75146577](https://avatars.discourse-cdn.com/v4/letter/a/f07891/32.png) [@anon75146577](https://discourse.mc-stan.org/u/anon75146577)\
**Post date:** [August 9, 2018, 3:01pm UTC](https://discourse.mc-stan.org/t/dynamic-panel-data-models-with-stan/5136/2 "2018-08-09T15:01:59Z")

</div>

Hi Jim,

Can you expand a bit on exactly what the model you want to fit it? Is it the one in the display?

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<div class="post-metadata">

**Author:** ![arya](https://yyz2.discourse-cdn.com/flex030/user_avatar/discourse.mc-stan.org/arya/32/1626_2.png) [@arya](https://discourse.mc-stan.org/u/arya)\
**Post date:** [August 9, 2018, 3:16pm UTC](https://discourse.mc-stan.org/t/dynamic-panel-data-models-with-stan/5136/3 "2018-08-09T15:16:30Z")

</div>

Haven’t done these models personally but I tried a little experiment since it was quick to do, and you were right I can’t recover the parameters.

```R
generate_panel_data <- function(delta, beta, x) {
  
  N <- length(x)
  u <- rnorm(1)
  y <- rep(NA, N)
  y[1] <- u + beta*x[1] + rnorm(1)
  for(n in 2:N) {
    y[n] <- u + delta*y[n-1] + beta*x[n] + rnorm(1)
  }
  
  tibble(y = y, u = u, lag_y = lag(y), x = x)
}

data <- generate_panel_data(1.2, 3.0, rnorm(20))

fit <- stan(file = "panel.stan",
            data = list(N = nrow(data),
                        x_lead = data$x %>% head(nrow(data)-1),
                        y_lead = data$y %>% head(nrow(data)-1),
                        y_lag = data$y %>% tail(nrow(data)-1)),
            chains = 1)

> fit
Inference for Stan model: panel.
1 chains, each with iter=2000; warmup=1000; thin=1; 
post-warmup draws per chain=1000, total post-warmup draws=1000.

        mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat
u -0.73 0.02 0.34 -1.38 -0.96 -0.74 -0.51 -0.08 431 1.00
delta 0.83 0.00 0.00 0.82 0.83 0.83 0.83 0.84 470 1.00
beta 0.45 0.01 0.26 -0.03 0.28 0.45 0.62 0.97 577 1.00
lp__ -48.11 0.07 1.30 -51.53 -48.70 -47.72 -47.18 -46.75 340 1.01

```

```stan
data {
  int N;
  vector[N-1] x_lead;
  vector[N-1] y_lead;
  vector[N-1] y_lag;
}
parameters {
  real u;
  real delta;
  real beta;
}
model {
  y_lead ~ normal(u + delta*y_lag + beta*x_lead, 1.0);
}

```

---

<div class="post-metadata">

**Author:** ![James\_Savage](https://yyz2.discourse-cdn.com/flex030/user_avatar/discourse.mc-stan.org/james_savage/32/26_2.png) [@James\_Savage](https://discourse.mc-stan.org/u/James_Savage)\
**Post date:** [August 9, 2018, 3:19pm UTC](https://discourse.mc-stan.org/t/dynamic-panel-data-models-with-stan/5136/4 "2018-08-09T15:19:46Z")

</div>

Yep @anon75146577 here’s some simulated data etc.

```

library(rstanarm); library(tidyverse)

# Simulate some fake data
delta <- 0.5
sigma_u <- 2
sigma_e <- 1
I <- 100
T <- 10

initial_values <- data_frame(individual = 1:I,
                             initial_y = rnorm(I),
                             u = rnorm(I, 0, sigma_u))

simulated_panel <- initial_values %>%
  group_by(individual) %>% 
  do({
    simulate <- function(x) {
      out <- data.frame(y = rep(NA, T),
                        time = rep(NA, T),
                        u = x$u)
      for(t in 1:T) {
        out$time[t] <- t
        if(t == 1) {
          out$y[t] <- rnorm(1, delta * x$initial_y + x$u, sigma_e)
        } else {
          out$y[t] <- rnorm(1, delta * out$y[t-1] + x$u, sigma_e)
        }
      }
      return(out)
    }
    as.data.frame(simulate(.))
  }) %>% 
  group_by(individual) %>% 
  mutate(lagged_y = lag(y)) %>% 
  filter(!is.na(lagged_y))

cor(simulated_panel$lagged_y, simulated_panel$u)

test_fit <- stan_lmer(y ~ lagged_y + (1 | individual), data = simulated_panel, iter = 1000, cores = 4)

bayesplot::mcmc_dens(as.data.frame(test_fit), pars = "lagged_y") +
  geom_vline(aes(xintercept = delta)) +
  annotate("text", y = 15, x = delta + 0.05, label = "true coef") +
  xlim(0, max(as.data.frame(test_fit)$lagged_y))

```

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<div class="post-metadata">

**Author:** ![arya](https://yyz2.discourse-cdn.com/flex030/user_avatar/discourse.mc-stan.org/arya/32/1626_2.png) [@arya](https://discourse.mc-stan.org/u/arya)\
**Post date:** [August 9, 2018, 3:26pm UTC](https://discourse.mc-stan.org/t/dynamic-panel-data-models-with-stan/5136/5 "2018-08-09T15:26:12Z")

</div>

Man I wish we had R syntax highlighting on the forums!

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<div class="post-metadata">

**Author:** ![James\_Savage](https://yyz2.discourse-cdn.com/flex030/user_avatar/discourse.mc-stan.org/james_savage/32/26_2.png) [@James\_Savage](https://discourse.mc-stan.org/u/James_Savage)\
**Post date:** [August 9, 2018, 4:07pm UTC](https://discourse.mc-stan.org/t/dynamic-panel-data-models-with-stan/5136/6 "2018-08-09T16:07:11Z")

</div>

> [@arya](#):
>
> generate\_panel\_data \<- function(delta, beta, x) { N \<- length(x) u \<- rnorm(1) y \<- rep(NA, N) y[1] \<- u + beta_x[1] + rnorm(1) for(n in 2:N) { y[n] \<- u + delta_y[n-1] + beta\*x[n] + rnorm(1) } tibble(y = y, u = u, lag\_y = lag(y), x = x) } data \<- generate\_panel\_data(1.2, 3.0, rnorm(20))

Arya – this isn’t repeated observations across multiple individuals it it?

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<div class="post-metadata">

**Author:** ![saudiwin](https://yyz2.discourse-cdn.com/flex030/user_avatar/discourse.mc-stan.org/saudiwin/32/1137_2.png) [@saudiwin](https://discourse.mc-stan.org/u/saudiwin)\
**Post date:** [August 9, 2018, 4:17pm UTC](https://discourse.mc-stan.org/t/dynamic-panel-data-models-with-stan/5136/7 "2018-08-09T16:17:29Z")

</div>

The simulation is recursive but the Stan program isn’t. Is that a problem?

---

<div class="post-metadata">

**Author:** ![arya](https://yyz2.discourse-cdn.com/flex030/user_avatar/discourse.mc-stan.org/arya/32/1626_2.png) [@arya](https://discourse.mc-stan.org/u/arya)\
**Post date:** [August 10, 2018, 12:22pm UTC](https://discourse.mc-stan.org/t/dynamic-panel-data-models-with-stan/5136/8 "2018-08-10T12:22:06Z")

</div>

No just a single individual, but without the \alpha term. I figured I’d start simple. The posterior pairs plots don’t actually look bad for that model or show any non-identifiabilities.

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<div class="post-metadata">

**Author:** ![James\_Savage](https://yyz2.discourse-cdn.com/flex030/user_avatar/discourse.mc-stan.org/james_savage/32/26_2.png) [@James\_Savage](https://discourse.mc-stan.org/u/James_Savage)\
**Post date:** [August 13, 2018, 8:29pm UTC](https://discourse.mc-stan.org/t/dynamic-panel-data-models-with-stan/5136/9 "2018-08-13T20:29:32Z")

</div>

Just bumping this in case the right person missed it. Seems too common a problem to not have been discussed on the forum before.

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<div class="post-metadata">

**Author:** ![Mauricio\_Garnier-Villarre](https://yyz2.discourse-cdn.com/flex030/user_avatar/discourse.mc-stan.org/mauricio_garnier-villarre/32/5702_2.png) [@Mauricio\_Garnier-Villarre](https://discourse.mc-stan.org/u/Mauricio_Garnier-Villarre)\
**Post date:** [August 14, 2018, 11:30pm UTC](https://discourse.mc-stan.org/t/dynamic-panel-data-models-with-stan/5136/10 "2018-08-14T23:30:28Z")

</div>

You should look at the R package ctsem, got continuous time SEM. This allows you to do dynamic systems with either frequentist or Bayesian, Bayesian is done with Stan.

The simplest version of the dinámica system is the panel model.

---

<div class="post-metadata">

**Author:** ![Charles\_Driver](https://yyz2.discourse-cdn.com/flex030/user_avatar/discourse.mc-stan.org/charles_driver/32/6524_2.png) [@Charles\_Driver](https://discourse.mc-stan.org/u/Charles_Driver)\
**Post date:** [August 15, 2018, 6:12pm UTC](https://discourse.mc-stan.org/t/dynamic-panel-data-models-with-stan/5136/11 "2018-08-15T18:12:23Z")

</div>

I never took the time to quite wrap my head around the Arellano Bond thing, but the poor performance of the rstanarm spec was eye opening. As Mauricio said, ctsem provides a front end for specifying hierarchical time series / state space models in Stan. I took a quick look and it seems to behave as you want, though performance is much slower than it could be for a perfectly observed first order model like you have (since there is no need to integrate out the latent states).

```
install.packages("devtools")
library(devtools)
install_github("cdriveraus/ctsem")

library(rstanarm); library(tidyverse); library(ctsem)

# Simulate some fake data
delta <- 0.5
sigma_u <- 2
sigma_e <- 1
I <- 100
T <- 10

initial_values <- data_frame(individual = 1:I,
                             initial_y = rnorm(I),
                             u = rnorm(I, 0, sigma_u))

simulated_panel <- initial_values %>%
  group_by(individual) %>% 
  do({
    simulate <- function(x) {
      out <- data.frame(y = rep(NA, T),
                        time = rep(NA, T),
                        u = x$u)
      for(t in 1:T) {
        out$time[t] <- t
        if(t == 1) {
          out$y[t] <- rnorm(1, delta * x$initial_y + x$u, sigma_e)
        } else {
          out$y[t] <- rnorm(1, delta * out$y[t-1] + x$u, sigma_e)
        }
      }
      return(out)
    }
    as.data.frame(simulate(.))
  }) %>% 
  group_by(individual) %>% 
  mutate(lagged_y = lag(y)) %>% 
  filter(!is.na(lagged_y))

cor(simulated_panel$lagged_y, simulated_panel$u)

#ctsem model
ctm <- ctModel(n.manifest = 1,n.latent = 1,LAMBDA = diag(1), type = 'standt',
  MANIFESTMEANS=matrix(0,1,1),CINT=matrix('u',1,1),
  MANIFESTVAR=diag(1e-6,1),
  T0VAR=matrix('stationary',1,1),
  manifestNames = 'y')

#set all parameters except the initial states and intercept fixed across subjects
ctm$pars$indvarying[!ctm$pars$matrix %in% c('T0MEANS','CINT')] <- FALSE

#adjust id column name
ctm$subjectIDname <- 'individual'

#fit using optimization
ctf <- ctStanFit(as.data.frame(simulated_panel),ctstanmodel = ctm,
  chains=6,optimize=TRUE,verbose=0,iter=300,deoptim = F,isloops = 0,issamples = 100)
summary(ctf)

#fit with sampling
ctf <- ctStanFit(as.data.frame(simulated_panel),ctstanmodel = ctm,chains=6,iter=300)

cat(ctf$stanmodeltext) #to view the stan model -- lots of unnecessary stuff for this problem though!
```

---

<div class="post-metadata">

**Author:** ![Bob\_Carpenter](https://yyz2.discourse-cdn.com/flex030/user_avatar/discourse.mc-stan.org/bob_carpenter/32/9230_2.png) [@Bob\_Carpenter](https://discourse.mc-stan.org/u/Bob_Carpenter)\
**Post date:** [August 22, 2018, 12:38pm UTC](https://discourse.mc-stan.org/t/dynamic-panel-data-models-with-stan/5136/12 "2018-08-22T12:38:16Z")

</div>

> [@arya](#):
>
> Man I wish we had R syntax highlighting on the forums!

Ask @syclik—somehow he or someone else turned on Stan highlighting. So I imagine if there’s a Stan plugin, there’s an R plugin!

---

<div class="post-metadata">

**Author:** ![rtrangucci](https://yyz2.discourse-cdn.com/flex030/user_avatar/discourse.mc-stan.org/rtrangucci/32/8375_2.png) [@rtrangucci](https://discourse.mc-stan.org/u/rtrangucci)\
**Post date:** [September 7, 2018, 12:19pm UTC](https://discourse.mc-stan.org/t/dynamic-panel-data-models-with-stan/5136/13 "2018-09-07T12:19:05Z")

</div>

Here’s my solution, with some code.

```
set.seed(123)
delta <- 0.5
sigma_u <- 0.5
sigma_e <- 1
I <- 100
T <- 30
u = rnorm(I, 0, sigma_u)
initial_values <- data_frame(
  individual = 1:I,
  u = u,
  initial_y = u + sigma_e / 
                  sqrt(1 - delta^2) * rnorm(I) # p(y_1 | u, sigma_e, delta)
)

simulated_panel <- initial_values %>%
  group_by(individual) %>% 
  do({
    simulate <- function(x) {
      out <- data.frame(y = rep(NA, T),
                        time = rep(NA, T),
                        u = x$u)
      for(t in 1:T) {
        out$time[t] <- t
        if(t == 1) {
          out$y[t] <- x$initial_y
        } else {
          out$y[t] <- rnorm(1, delta * (out$y[t-1] - x$u) + x$u, sigma_e)
        }
      }
      return(out)
    }
    as.data.frame(simulate(.))
  }) %>% 
  group_by(individual) %>% 
  mutate(lagged_y = lag(y)) %>% ungroup()

y <- matrix(0, T, I)
y_lag <- matrix(0, T, I)
for (i in 1:I) {
  y[,i] <- filter(simulated_panel, individual == i)$y
  y_lag[2:T,i] <- filter(simulated_panel, individual == i)$lagged_y[2:T]
}

stan_dat <- list(I = I, T = T, y = y, y_lag = y_lag)
library(rstan)
mod <- stan_model('dynamic-panel.stan')
fit <- sampling(mod, data = stan_dat, cores = 4, chains = 4, iter = 2000)

```

Here’s `dynamic-panel.stan`:

```
data {
  int<lower=1> I;
  int<lower=1> T;
  matrix[T,I] y;
  matrix[T,I] y_lag;
}
parameters {
  real<lower=0,upper=1> delta_raw;
  real<lower=0> sigma_u;
  real<lower=0> sigma_e;
  vector[I] u_raw;
}
transformed parameters {
  vector[I] u = sigma_u * u_raw;
  real delta = delta_raw * 2 - 1;
}
model {
  real one_minus_delta_sq = (1 - delta) * (1 + delta);
  vector[I] u_delta = u * (1 - delta);
  for (i in 1:I) {
    target += normal_lpdf(y[1,i] | u[i], 
                          sigma_e / sqrt(one_minus_delta_sq));
    target += normal_lpdf(y[2:T,i] | u_delta[i]
                                     + delta * y_lag[2:T,i], sigma_e);
  }
  u_raw ~ normal(0, 1);
  sigma_u ~ normal(0, 2);
  sigma_e ~ normal(0, 1);
  delta_raw ~ beta(4, 4);
}

```

Summary of the model run over the simulated data from above:

```
> print(fit , pars = c('delta','sigma_u','sigma_e'))
Inference for Stan model: dynamic-panel.
4 chains, each with iter=2000; warmup=1000; thin=1; 
post-warmup draws per chain=1000, total post-warmup draws=4000.

        mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat
delta 0.50 0.00 0.04 0.42 0.47 0.50 0.52 0.57 831 1.01
sigma_u 0.36 0.01 0.13 0.05 0.29 0.37 0.45 0.58 430 1.01
sigma_e 1.01 0.00 0.02 0.96 0.99 1.01 1.03 1.06 4000 1.00

Samples were drawn using NUTS(diag_e) at Fri Sep 7 08:03:36 2018.
For each parameter, n_eff is a crude measure of effective sample size,
and Rhat is the potential scale reduction factor on split chains (at 
convergence, Rhat=1).

```

When I boost the number of observations per individual to 50 the inference gets sharper for `sigma_u` and approaches the true value.

```
Inference for Stan model: dynamic-panel.
4 chains, each with iter=2000; warmup=1000; thin=1; 
post-warmup draws per chain=1000, total post-warmup draws=4000.

        mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat
delta 0.48 0 0.01 0.46 0.47 0.48 0.49 0.51 4000 1
sigma_u 0.46 0 0.04 0.38 0.43 0.46 0.49 0.55 1400 1
sigma_e 0.99 0 0.01 0.98 0.99 0.99 1.00 1.01 4000 1

Samples were drawn using NUTS(diag_e) at Fri Sep 7 08:17:52 2018.
For each parameter, n_eff is a crude measure of effective sample size,
and Rhat is the potential scale reduction factor on split chains (at 
convergence, Rhat=1).

```

---

<div class="post-metadata">

**Author:** ![Saram\_Han](https://avatars.discourse-cdn.com/v4/letter/s/76d3ee/32.png) [@Saram\_Han](https://discourse.mc-stan.org/u/Saram_Han)\
**Post date:** [January 8, 2019, 6:16pm UTC](https://discourse.mc-stan.org/t/dynamic-panel-data-models-with-stan/5136/14 "2019-01-08T18:16:18Z")

</div>

Could you refer the paper or model that you based on to write your code? I would like to use your model but I can’t find the dynamic panel data model that you are using in your model.  
EDIT:  
Just in case if someone is interested in this approach who has no econometric background, I found the following paper useful to understand rtrangucci’s code.  
[https://www.researchgate.net/profile/Hashem\_Pesaran/publication/226673422\_Random\_Coefficient\_Models/links/0fcfd5089366b8067f000000/Random-Coefficient-Models.pdf](https://www.researchgate.net/profile/Hashem_Pesaran/publication/226673422_Random_Coefficient_Models/links/0fcfd5089366b8067f000000/Random-Coefficient-Models.pdf)

---

<div class="post-metadata">

**Author:** ![c5v](https://avatars.discourse-cdn.com/v4/letter/c/e19adc/32.png) [@c5v](https://discourse.mc-stan.org/u/c5v)\
**Post date:** [January 30, 2019, 1:41pm UTC](https://discourse.mc-stan.org/t/dynamic-panel-data-models-with-stan/5136/16 "2019-01-30T13:41:18Z")

</div>

Could you maybe explain why you multiplicate by `1-delta` in `vector[I] u_delta = u * (1 - delta);`?  
Why don’t you just use `u[i]` instead of `u_delta[i]` in `target += normal_lpdf(y[2:T,i] | u_delta[i] + delta * y_lag[2:T,i], sigma_e);`?

---

<div class="post-metadata">

**Author:** ![rtrangucci](https://yyz2.discourse-cdn.com/flex030/user_avatar/discourse.mc-stan.org/rtrangucci/32/8375_2.png) [@rtrangucci](https://discourse.mc-stan.org/u/rtrangucci)\
**Post date:** [January 31, 2019, 6:40pm UTC](https://discourse.mc-stan.org/t/dynamic-panel-data-models-with-stan/5136/17 "2019-01-31T18:40:43Z")

</div>

Hi @c5v, sorry for the delay. The use of `u_delta[i]` instead of `u[i]` is just an algebraic trick to cut down on computation. We could instead write the likelihood like:

`target += normal_lpdf(y[2:T,i] | u[i] + delta * (y_lag[2:T,i] - u[i]), sigma_e)`

which leads to

`target += normal_lpdf(y[2:T,i] | u[i] + delta * y_lag[2:T,i] - delta * u[i], sigma_e)`

simplifying to

`target += normal_lpdf(y[2:T,i] | u[i] * (1 - delta) + delta * y_lag[2:T,i], sigma_e)`

---

<div class="post-metadata">

**Author:** ![c5v](https://avatars.discourse-cdn.com/v4/letter/c/e19adc/32.png) [@c5v](https://discourse.mc-stan.org/u/c5v)\
**Post date:** [January 31, 2019, 6:53pm UTC](https://discourse.mc-stan.org/t/dynamic-panel-data-models-with-stan/5136/18 "2019-01-31T18:53:27Z")

</div>

@rtrangucci Thanks for your reply, this clarifies a lot!

The only thing I don’t really get is why you subtract `u[i]` in `delta*(y_lag[2:T,i] - u[i])` since we are not modelling first differences here, right? Could you maybe explain this to me, this will help me a lot!

---

<div class="post-metadata">

**Author:** ![rtrangucci](https://yyz2.discourse-cdn.com/flex030/user_avatar/discourse.mc-stan.org/rtrangucci/32/8375_2.png) [@rtrangucci](https://discourse.mc-stan.org/u/rtrangucci)\
**Post date:** [January 31, 2019, 7:10pm UTC](https://discourse.mc-stan.org/t/dynamic-panel-data-models-with-stan/5136/19 "2019-01-31T19:10:21Z")

</div>

It’s because I want the mean of each observation in group `i` to be `u[i]`. The model is

Y\_{i,t} = \mu\_i + \delta (Y\_{i,t-1} - \mu\_i) + \epsilon\_{i,t}\\ \epsilon\_{i,t} \sim \text{Normal}(0,1)

Taking expectations of each side:

\mathbb{E}[Y\_{i,t}] = \mu\_i + \delta \mathbb{E}[(Y\_{i,t-1} - \mu\_i)] \\ \mathbb{E}[Y\_{i,t}] = \mu\_i + \delta (\mathbb{E}[Y\_{i,t-1}] - \mu\_i) \\ \mathbb{E}[Y\_{i,t}] = \mu\_i

---

<div class="post-metadata">

**Author:** ![c5v](https://avatars.discourse-cdn.com/v4/letter/c/e19adc/32.png) [@c5v](https://discourse.mc-stan.org/u/c5v)\
**Post date:** [January 31, 2019, 7:19pm UTC](https://discourse.mc-stan.org/t/dynamic-panel-data-models-with-stan/5136/20 "2019-01-31T19:19:57Z")

</div>

@rtrangucci clear explanation thanks!

If I am right, in a multilevel setting (which I am building right now), which then looks like y\_{ijt} = \beta\_{0ij} + \beta\_{1ij}x\_{ijt} + \delta y\_{ij,t-1} + \epsilon\_{ijt}, I could also use your reasoning and model it like:

y\_{ijt} = \mu\_{ij} + \delta(y\_{ij,t-1} - \mu\_{ij}) + \epsilon\_{ijt}, where \mu\_{ij} = \beta\_{0ij} + \beta\_{1ij}x\_{ijt}

and use your approach, right?

Sorry for all my questions, but I am really stuck with this problem.

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<div class="post-metadata">

**Author:** ![rtrangucci](https://yyz2.discourse-cdn.com/flex030/user_avatar/discourse.mc-stan.org/rtrangucci/32/8375_2.png) [@rtrangucci](https://discourse.mc-stan.org/u/rtrangucci)\
**Post date:** [January 31, 2019, 7:21pm UTC](https://discourse.mc-stan.org/t/dynamic-panel-data-models-with-stan/5136/21 "2019-01-31T19:21:59Z")

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No problem! You’re correct, you could model your process like that, and you’d get the nice property that \mathbb{E}[y\_{ijt}] = \mu\_{ij}

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