Examples from Lopes and Tsay (2011)

Example 1: Local level model – state filtering

Comparison of four particle filters for the local level model. For \(t = 1,\ldots,n\), the local level model can be written as \[\begin{eqnarray*} y_t &\sim& N(x_t,\sigma^2)\\ x_t &\sim& N(\alpha x_{t-1},\tau^2), \end{eqnarray*}\] with initial value \(x_0 \sim N(m_0,C_0)\)and known quantities \((\alpha,\sigma^2,\tau^2,m_0,C_0)\). The four particle filters are

1. bootstrap filter (Gordon et al, 1993),

2. auxiliary particle filter (Pitt and Shephard, 1999),

3. optimal bootstrap filter, and

4. optimal auxiliary particle filter.

R code: https://hedibert.org/wp-content/uploads/2014/06/Example1-locallevelmodel-R.txt

Example 4: Local level model – state filtering & parameter learning

Comparison of three particle filters for the local level model when parameter learning is also taken into account. For t=1,,n, the model can be written as \[\begin{eqnarray*} y_t|x_t &\sim& N(x_t,\sigma^2)\\ x_t|x_{t-1} &\sim& N(\alpha+\beta x_{t-1},\tau^2) \end{eqnarray*}\] with additional priors \[\begin{eqnarray*} x_0 &\sim& N(m_0,C_0)\\ \sigma^2 &\sim& IG(a_0,A_0)\\ \alpha|\tau^2 &\sim& N(b_{01},\tau^2B_{01})\\ \beta|\tau^2 &\sim& N(b_{02},\tau^2B_{02})\\ \tau^2 &\sim& IG(\nu_0/2,\nu_0\tau^2_0/2), \end{eqnarray*}\] and known hyperparameters \((m_0,C_0)\), \((a_0,A_0)\), \((b_0,B_0)\) and \((\nu_0,\tau^2_0)\). The three filters are

1. Liu and West filter (Liu and West, 2001),

2. Storvik filter (Storvik, 2002), and

3. Particle learning (Carvalho et al., 2010)

R code: http://hedibert.org/wp-content/uploads/2014/06/Example4-comparison-lw-sf-pl-R.txt

Example 4: Local level model – PL versus MCMC

Same context as the above example:

1. Comparison of MCMC and optimal auxiliary particle filter,

2. Comparison of MCMC and particle learning.

R code: http://hedibert.org/wp-content/uploads/2014/06/Example4-comparison-pl-mcmc-R.txt

Application 1: Dynamic Beta regression

Brazilian monthly unemployment rate from March 2002 to December 2009 (IBGE).

R code: http://hedibert.org/wp-content/uploads/2014/06/Application1-dynamicbetaregression-R.txt

Application 2: Stochastic volatility model

For \(t=1,\cdots,n\), the basic normal stochastic volatility model can be written as \[\begin{eqnarray*} y_t | x_t &\sim& N(0, \exp\{x_t/2\})\\ x_t | x_{t-1} &\sim& N(\alpha + \beta x_{t-1}, \tau^2) \end{eqnarray*}\] with \(x_0 \sim N(m_0,C_0)\), \(\alpha|\tau^2 \sim N(b_{01},\tau^2 B_{01})\), \(\beta|\tau^2 \sim N(b_{02},\tau^2 B_{02})\), where \(\tau^2 \sim IG(\nu_0/2,\nu_0\tau^2_0/2)\) and known hyper-parameters \(m_0\), \(C_0\), \(B_0\), \(\nu_0\) and \(\tau_0^2\).

Data: Monthly log returns of GE stock.

Period: January 1926 to December 1999 (888 observations).

Source: Tsay (2005), Chapter 12, Example 12.6, page 591.

Dataset: https://www.chicagobooth.edu/-/media/faculty/ruey-s-tsay/teaching/fts2/m-geln.txt

R code: http://hedibert.org/wp-content/uploads/2014/06/Application2-stochasticvolatility-R.txt

Application 3: Realized volatility

We entertain two realized volatility models:

1. Three RV time series are modeled by independent univariate local level models

2. Trivariate vector of RV time series is modeled by a multivariate local level.

Data: Intradaily realized volatility of Alcoa stock (5m, 10m, 20m) from 2 January 2003 to 7 May 2004 for 340 observations. The daily realized volatilities used are the the sums of squares of intraday 5 min, 10 min and 20 min log returns measured in percentages. Tsay (2005), Chapter 11: State-Space Models and Kalman Filter.

Dataset: https://www.chicagobooth.edu/-/media/faculty/ruey-s-tsay/teaching/fts2/aa-3rv.txt

R code: http://hedibert.org/wp-content/uploads/2014/06/Application3-realizedvolatility-R.txt