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bayesian-stochastic-volatility.Rmd
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bayesian-stochastic-volatility.Rmd
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# Bayesian Stochastic Volatility Model
Stochastic Volatility Model for centered time series over $t$ equally spaced
points. The latent parameter $h$ is the log volatility, φ the persistence of the
volatility and μ the mean log volatility. ϵ is the white-noise shock and δ the
shock on volatility. The Stan code is based on that in the manual (at the time
I originally played with it).
y_t = exp(h_t/2)*ϵ_t
h_t = μ + φ\*(h_{t-1}-μ) + δ_t*σ
h_1 ~ N(μ, σ/sqrt(1-φ^2))
ϵ_t ~ N(0,1); δ_t ~ N(0,1)
With some rearranging:
ϵ_t = y_t*exp(-h_t/2)
y_t ~ N(0, exp(h_t/2)
h_t ~ N(μ + φ*(h_t-μ), σ)
## Data Setup
The data regards inflation based on the U.S. consumer price index (`inflation = 400*log(cpi_t/cpi_{t-1}`), from the second quarter of 1947 to the second quarter
of 2011 (from Statistical Computation and Modeling 2014, chap 11).
```{r bayes-sv-setup}
library(tidyverse)
d = read_csv(
'https://raw.githubusercontent.com/m-clark/Datasets/master/us%20cpi/USCPI.csv',
col_names = 'inflation'
)
inflation = pull(d, inflation)
summary(inflation)
inflation_cen = scale(inflation, scale = FALSE)
```
## Model Code
This original code keeps to the above formulation but can take a long time to converge. ϵ_t and δ_t are implicit.
```{stan bayes-sv1, output.var = 'bayes_conceptual'}
data {
int<lower = 0> N_t; // Number of time points (equally spaced)
vector[N_t] y; // mean corrected response at time t
}
parameters {
real mu; // mean log volatility
real<lower = -1,upper = 1> phi; // persistence of volatility
real<lower = 0> sigma; // white noise shock scale
vector[N_t] h; // log volatility at time t
}
model {
//priors
phi ~ uniform(-1, 1);
sigma ~ cauchy(0, 5);
mu ~ cauchy(0, 10);
//likelihood
h[1] ~ normal(mu, sigma / sqrt(1 - phi * phi));
for (t in 2:N_t)
h[t] ~ normal(mu + phi * (h[t - 1] - mu), sigma);
for (t in 1:N_t)
y ~ normal(0, exp(h[t] / 2));
}
```
This code is more performant and will be used to actually estimate the model.
```{stan bayes-sv-2, output.var='bayes_sv'}
data {
int<lower = 0> N_t; // N time points (equally spaced)
vector[N_t] y; // mean corrected response at time t
}
parameters {
real mu; // mean log volatility
real<lower = -1,upper = 1> phi; // persistence of volatility
real<lower = 0> sigma; // white noise shock scale
vector[N_t] h_std; // standardized log volatility at time t
}
transformed parameters{
vector[N_t] h; // log volatility at time t
h = h_std * sigma;
h[1] = h[1] / sqrt(1-phi * phi);
h = h + mu;
for (t in 2:N_t)
h[t] = h[t] + phi * (h[t-1] - mu);
}
model {
//priors
phi ~ uniform(-1, 1);
sigma ~ cauchy(0, 5);
mu ~ cauchy(0, 10);
h_std ~ normal(0, 1);
//likelihood
y ~ normal(0, exp(h/2));
}
generated quantities{
vector[N_t] y_rep;
for (t in 1:N_t){
y_rep[t] = normal_rng(0, exp(h[t]/2));
}
}
```
## Estimation
We can use `c()` to get rid of matrix format, or specify as matrix instead of vector in model code.
```{r bayes-sv-est, results='hide'}
stan_data = list(N_t = length(inflation_cen), y = c(inflation_cen))
library(rstan)
fit = sampling(
bayes_sv,
data = stan_data,
cores = 4,
thin = 4
)
```
## Results
Explore the results.
```{r bayes-sv-result}
print(
fit,
digits = 3,
par = c('mu', 'phi', 'sigma'),
probs = c(.025, .5, .975)
)
```
## Visualization
With the necessary components in place, we can visualize our predictions. Compare to fig. 11.1 in the text.
```{r bayes-sv-vis1}
# Create y_rep 'by-hand'
h = extract(fit, 'h')$h
# y_rep = apply(h, 1, function(h) rnorm(length(inflation), 0, exp(h / 2)))
# or just extract
y_rep = extract(fit, 'y_rep')$y_rep
h = colMeans(h)
library(lubridate)
library(scales)
series = ymd(paste0(rep(1947:2014, e = 4), '-', c('01', '04', '07', '10'), '-', '01'))
seriestext = series[1:length(inflation)]
```
```{r bayes-sv-vis2, echo=FALSE}
qplot(
seriestext,
h,
width = .5,
color = I('gray50'),
geom = 'line',
ylab = 'log volatility'
)
gdat0 = tibble(
date = as_date(seriestext),
inflation = inflation_cen[,1]
)
gdat_sim = t(y_rep[sample(1:ncol(y_rep), 10), ]) %>%
as.data.frame() %>%
mutate(date = gdat0$date) %>%
pivot_longer(-date, names_to = 'iter', values_to = 'inflation')
gdat0 %>%
ggplot(aes(date, inflation)) +
geom_line(aes(group = iter),
color = '#ff5500',
alpha = .1,
data = gdat_sim) +
geom_line(color = 'gray25') +
labs(y = 'Inflation\ncentered', caption = '10 posterior predictive draws shown in color')
```
## Source
Original code available at:
https://github.com/m-clark/Miscellaneous-R-Code/blob/master/ModelFitting/Bayesian/stochasticVolatility.R