time-series-models

# Time Series Models

## AR(1) Model

### Stan
```stan
data {
  int<lower=0> T;
  vector[T] y;
}
parameters {
  real mu;
  real<lower=-1, upper=1> phi;  // Stationarity
  real<lower=0> sigma;
}
model {
  mu ~ normal(0, 10);
  phi ~ uniform(-1, 1);
  sigma ~ exponential(1);

  // Stationary initial distribution
  y[1] ~ normal(mu, sigma / sqrt(1 - phi^2));

  // AR(1) likelihood
  for (t in 2:T)
    y[t] ~ normal(mu + phi * (y[t-1] - mu), sigma);
}
```

### Vectorized Stan (Efficient)
```stan
model {
  y[1] ~ normal(mu, sigma / sqrt(1 - square(phi)));
  y[2:T] ~ normal(mu + phi * (y[1:(T-1)] - mu), sigma);
}
```

### JAGS
```
model {
  y[1] ~ dnorm(mu, tau / (1 - phi * phi))
  for (t in 2:T) {
    y[t] ~ dnorm(mu + phi * (y[t-1] - mu), tau)
  }
  mu ~ dnorm(0, 0.001)
  phi ~ dunif(-1, 1)
  tau ~ dgamma(0.001, 0.001)
  sigma <- 1/sqrt(tau)
}
```

## AR(p) Model

### Stan
```stan
data {
  int<lower=0> T;
  int<lower=1> P;  // AR order
  vector[T] y;
}
parameters {
  real mu;
  vector[P] phi;
  real<lower=0> sigma;
}
model {
  mu ~ normal(0, 10);
  phi ~ normal(0, 0.5);
  sigma ~ exponential(1);

  for (t in (P+1):T) {
    real pred = mu;
    for (p in 1:P)
      pred += phi[p] * (y[t-p] - mu);
    y[t] ~ normal(pred, sigma);
  }
}
```

## Local Level (Random Walk + Noise)

### Stan
```stan
data {
  int<lower=0> T;
  vector[T] y;
}
parameters {
  vector[T] mu;           // Latent state
  real<lower=0> sigma_y;  // Observation noise
  real<lower=0> sigma_mu; // State noise
}
model {
  sigma_y ~ exponential(1);
  sigma_mu ~ exponential(1);

  // State evolution (random walk)
  mu[1] ~ normal(y[1], sigma_y);
  mu[2:T] ~ normal(mu[1:(T-1)], sigma_mu);

  // Observations
  y ~ normal(mu, sigma_y);
}
```

## Local Linear Trend

### Stan
```stan
parameters {
  vector[T] mu;           // Level
  vector[T] delta;        // Trend
  real<lower=0> sigma_y;
  real<lower=0> sigma_mu;
  real<lower=0> sigma_delta;
}
model {
  // Level evolution
  mu[2:T] ~ normal(mu[1:(T-1)] + delta[1: