MATH11175 Bayesian Data Analysis - Assignment 1: Stochastic Volatility (SV) model and predictive model
Bayesian Data Analysis, 2022/2023, Semester 2
Assignment 1
IMPORTANT INFORMATION ABOUT THE ASSIGNMENT
In this paragraph, we summarize the essential information about this assignment. The format and rules for this assignment are different from your other courses, so please pay attention. 1) Deadline: The deadline for submitting your solutions to this assignment is the 6 March 12:00 noon Edinburgh time. 2) Format: You will need to submit your work as 2 components: a PDF report, and your R Markdown (.Rmd) notebook. There will be two separate submission systems on Learn:
Gradescope for the report in PDF format, and a Learn assignment for the code in Rmd format. You need to write your solutions into this R Markdown notebook (code in R chunks and explanations in Markdown chunks), and then select Knit/Knit to PDF in RStudio to create a PDF report.
The compiled PDF needs to contain everything in this notebook, with your code sections clearly visible (not hidden), and the output of your code included. Reports without the code displayed in the PDF, or without the output of your code included in the PDF will be marked as 0, with the only feedback “Report did not meet submission requirements”.
You need to upload this PDF in Gradescope submission system, and your Rmd file in the Learn assignment submission system. You will be required to tag every sub question on Gradescope. Some key points that are different from other courses:
Problem 1
In this problem, we study a dataset about currency exchange rates. The exrates dataset of the stochvol package contains the daily average exchange rates of 24 currencies versus the EUR, from 2000-01-03 until 2012-04-04.
require(stochvol)
## Loading required package: stochvol
data("exrates")
#You may need to set the working directory first before loading the dataset
#setwd("location of Assignment 1")
#The first 6 rows of the dataframe
print.data.frame(exrates[1:6,])
## AUD CAD CHF CZK DKK GBP HKD IDR JPY
## 2000/01/03 1.5346 1.4577 1.6043 36.063 7.4404 0.6246 7.8624 7133.32 102.75
## 2000/01/04 1.5677 1.4936 1.6053 36.270 7.4429 0.6296 8.0201 7265.16 105.88
## 2000/01/05 1.5773 1.5065 1.6060 36.337 7.4444 0.6324 8.0629 7437.97 107.34
## 2000/01/06 1.5828 1.5091 1.6068 36.243 7.4441 0.6302 8.0843 7495.52 108.72
## 2000/01/07 1.5738 1.5010 1.6079 36.027 7.4436 0.6262 8.0030 7398.88 108.09
## 2000/01/10 1.5587 1.4873 1.6089 35.988 7.4448 0.6249 7.9471 7320.49 107.26
## KRW MXN MYR NOK NZD PHP PLN RON RUB
## 2000/01/03 1140.02 9.6105 3.8422 8.0620 1.9331 40.424 4.1835 1.8273 27.7548
## 2000/01/04 1157.32 9.7453 3.9188 8.1500 1.9745 40.992 4.2423 1.8858 28.3594
## 2000/01/05 1176.08 9.8969 3.9393 8.2060 1.9956 41.637 4.2627 1.8979 28.3006
## 2000/01/06 1191.90 9.9751 3.9498 8.2030 2.0064 42.148 4.2593 1.9000 28.5111
## 2000/01/07 1169.52 9.8368 3.9100 8.1945 1.9942 41.493 4.1897 1.8822 28.2526
## 2000/01/10 1157.84 9.6449 3.8824 8.1900 1.9783 41.226 4.1567 1.8729 28.7609
## SEK SGD THB TRY USD date
## 2000/01/03 8.5520 1.6769 37.2793 0.546131 1.0090 2000-01-03
## 2000/01/04 8.6215 1.7047 38.2078 0.552354 1.0305 2000-01-04
## 2000/01/05 8.6415 1.7159 38.5375 0.555329 1.0368 2000-01-05
## 2000/01/06 8.6445 1.7291 39.0734 0.555674 1.0388 2000-01-06
## 2000/01/07 8.6450 1.7096 38.4299 0.554980 1.0284 2000-01-07
## 2000/01/10 8.6570 1.6975 38.0328 0.552469 1.0229 2000-01-10
cat(paste("Data from ", min(exrates$date)," until ",max(exrates$date)))
## Data from 2000-01-03 until 2012-04-04
As we can see, not all dates are included in the dataset. Some are missing, such as weekends, and public holidays.
In this problem, we are going to fit a various stochastic volatility models on this dataset (see e.g. https://www.jstor.org/stable/1392251). a)[10 marks] Consider the following leveraged Stochastic Volatility (SV) model. yt = β0 + β1 yt−1 + exp(ht /2)t for 1 ≤ t ≤ T, ht+1 = µ + φ(ht − µ) + σηt (t , ηt ) ∼ N (0, Σρ ) for for 0 ≤ t ≤ T, 1 ρΣρ =ρ 1h0 ∼ N (µ, σ 2 /(1 − φ2 )),
Here t is the time index, yt are the observations (such as daily USD/EUR rate), ht are the logvariance process, t is the observation noise, and ηt is the log-variance process noise (which are correlated, but independent for different values of $t$). The hyperparameters are β0 , β1 , µ, φ, σ, ρ.
For stability, it is necessary to have φ ∈ (−1, 1), and by the definition of correlation matrices, we have ρ ∈ [−1, 1].
Implement this model in JAGS or Stan on the first 3 months of USD/EUR data from the dataset, i.e. from dates 2000-01-03 until 2000-04-02.
Explain how did you choose priors for all parameters. Explain how did you take into account the days without observation in your model. Fit the model, do convergence diagnostics, print out the summary of the results, and discuss them.
Make sure that the Effective Sample Size is at least 1000 for all 6 hyperparameters (you need to choose burn-in and number of steps appropriately for this).
Explanation: (Write your explanation here) b)[10 marks] In practice, one often encounters outliers in exchange rates. These can be sometimes modeled by assuming Student’s t distribution in the observation errors (i.e. t ). The robust leveraged SV model can be expressed as yt = β0 + β1 yt−1 + exp(ht /2)t ht+1 = µ + φ(ht − µ) + σηt
for 1 ≤ t ≤ T,
for 0 ≤ t ≤ T,
h0 ∼ N (µ, σ 2 /(1 − φ2 )),
ηt ∼ N (0, 1) t |ηt ∼ tν (ρηt , 1). Here ν is the degrees of freedom parameter (unknown). Implement this model in JAGS or Stan on the first 3 months of USD/EUR data from the dataset.
Explain how did you choose priors for all parameters. Explain how did you take into account the days without observation in your model.
Fit the model, do convergence diagnostics, print out the summary of the results, and discuss them.
Make sure that the Effective Sample Size is at least 1000 for all 6 hyperparameters (you need to choose burn-in and number of steps appropriately for this). Explanation: (Write your explanation here) c)[10 marks] Perform posterior predictive checks on both models a) and b). Explain how did you choose the test functions. Discuss the results. Explanation: (Write your explanation here) d)[10 marks] Based on your models a) and b), plot the posterior predictive densities of the USD/EUR rate on the dates 2000-04-03, 2020-04-04 and 2020-04-05 (the next 3 days after the period considered). Compute the posterior means and 95% credible intervals. Discuss the results. Explanation: (Write your explanation here) e)[10 marks] In this question, we are going to look use a multivariate stochastic volatility model with leverage to study the USD/EUR and GBP/EUR exchange rates jointly. The model is described as follows, y t = β 0 + β 1 y t−1 + exp(ht /2)t ht+1 = φ(ht ) + η t
for 0 ≤ t ≤ T,
for 1 ≤ t ≤ T, h0 ∼ N (0, I),
(t , ηt ) ∼ N (0, Σ) . Here I denotes the 2 x 2 identity matrix, y t , β 0 , ht , η t , t are 2 dimensional vectors, β 1 and φ are 2 x 2 matrices, Σ is a 4 x 4 covariance matrix. At each time step t, the two components of yt will be used to model the USD/EUR and GBP/EUR exchange rates, respectively. Implement this model in JAGS or Stan. Discuss your choices for priors for every parameter Fit the model, do convergence diagnostics, print out the summary of the results, and discuss them. Explanation: (Write your explanation here)
Problem 2 - NBA data
In this problem, we are going to construct a predictive model for NBA games. We start by loading the dataset. games<-read.csv("games.csv") teams<-read.csv("teams.csv") games.csv contains the information about games such as GAME_DATE, SEASON, HOME_TEAM_ID, VISITOR_TEAM_ID, PTS_home (final score for home team) and PTS_away (final score for away team). teams.csv contains the names of each team, i.e. the names corresponding to each team ID. We are going to fit some Bayesian linear regression models on the scores of each team. You can use either INLA, JAGS or Stan. a)[10 marks] The dataset contains data from 20 seasons, but we are going to focus on only one, the 2021 season. Please only keep games where SEASON is 2021 in the dataset, and remove all other seasons. Please order the games according to the date of occurrence (they are not ordered like that in the dataset). The scores are going to be assumed to follow a linear Gaussian model, 2 SgH ∼ N (µHg , σ ),2gA ∼ N (µA , σ ).
Here SgH denotes the final score of the home team in game g, and SgA denotes the final score of the away team in game g.
Note that the true scores can only take non-negative integer values, so the Gaussian distribution is not perfect, but it can still be used nevertheless.
The means for the scores are going to be modeled as a combination of three terms: attacking strength, defending ability, and whether the team is playing at home, or away. For each team, we denote their attacking strength parameter by ateam , their defending strength parameter by dteam , and the effect of playing at home as h. This quantifies the effect of playing at home on the expected number of goals scored. Our model is the following (µH g is for the goals scored by the home team, and is µA is for the away team): gµHg = β0 + ahome.team + daway.team + hµA g = β0 + aaway.team + dhome.team
Implement this model. Select your own prior distributions for the parameters, and discuss the reason for using those priors.
Obtain the summary statistics for the posterior distribution of the model parameters. Evaluate the root mean square error (RMSE) of your posterior means versus the true scores. Interpret the results.
Explanation: (Write your explanation here) b)[10 marks] In part a), the model assumed that the home effect is the same for each team. In this part, we consider a team-specific home effect hhome.team , µH g = β0 + ahome.team + daway.team + hhome.team µA g = β0 + aaway.team + dhome.team Implement this model. Select your own prior distributions for the parameters, and discuss the reason for using those priors. Obtain the summary statistics for the posterior distribution of the model parameters. Evaluate the root mean square error (RMSE) of your posterior means versus the true scores. Interpret the results. Explanation: (Write your explanation here) c)[10 marks] Propose an improved linear model using the information in the dataset before the game (you cannot use any information in the same row as the game, as this is only available after the game). Hint: you can try incorporating running averages of some covariates specific to each team, by doing some pre-processing. Implement your model. Select your own prior distributions for the parameters, and discuss the reason for using those priors. Obtain the summary statistics for the posterior distribution of the model parameters.
Evaluate the root mean square error (RMSE) of your posterior means versus the true scores. Interpret the results. Explanation: (Write your explanation here) d)[10 marks] Perform posterior predictive checks on all 3 models a), b), and c). Explain how did you choose the test functions.
Discuss the results. Explanation: (Write your explanation here) e)[10 marks] In the previous questions, we were assuming a model of the form. 2 SgH ∼ N (µHg , σ ),2SgA ∼ N (µAg , σ ). It is natural to model these two results jointly with a multivariate normal, (SgH , SgA )
Implement such a model. The definition of µH g and µg can be either one of a), b), or c), you just need to implement one of them.
Explain how did you choose the prior on Σ Obtain the summary statistics for the posterior distribution of the model parameters. Evaluate the root mean square error (RMSE) of your posterior means versus the true scores. Interpret the results. Explanation: (Write your explanation here)
