STA2570 Winter 2023
Assignment 2
March 11, 2023
1. In this exercise we estimate a time series model using a Bayesian approach. We consider
US quarterly inflation data in inflation.csv; it contains 252 values. Consider the AR(2)
model
Yt = α+ φ1Yt?1 + φ2Yt?2 + t, t = 1, . . . , T,
where the t are i.i.d. N(0, σ
2
). Here the parameter is θ = (α, φ1, φ2, σ). We assume that
Y0 and Y?1 are given as known. So for our data set T = 250.
Suppose we use the following prior:
β = (α, φ1, φ2) ～ N(0, I3).
σ2 ～ InverseGamma.
Also β and σ2 are independent under the prior. The definition of the inverse gamma distri-
bution can be found here: https://en.wikipedia.org/wiki/Inverse-gamma_distribution
(a) Write down the likelihood, i.e., the joint density of (Y1, . . . , YT ) as a function of the
parameters.
(b) Consider the posterior density pi(θ|y), where y = (y1, . . . , yT ) is the observed series.
Derive carefully:
– The conditional distribution of β given 2 . (It is normal.)
– The conditional distribution of σ2 given β. (It is inverse gamma.)
(c) Implement the Gibb’s sampler using the results in (a). Report the posterior mean of
each parameter. Compare your results with the frequentist estimates (e.g. using the
function arima() in R).
References: Chapter 1 (in particular Sections 1.3.2–1.3.4) of Applied Bayesian Econometrics for Central
Bankers, available at
www.bankofengland.co.uk/ccbs/applied-bayesian-econometrics-for-central-bankers-updated-2017.
Also see this blog post which explains the material in a more user-friendly way:
towardsdatascience.com/a-bayesian-approach-to-time-series-forecasting-d97dd4168cb7.
2. Consider the following stocks (in the Dow Jones Index). In R, their tickers are given as
follows:
1
symbol_seq <- c("MMM", "AXP", "AMGN", "AAPL", "BA",
"CAT", "CVX", "CSCO", "KO", "DIS",
"GS", "HD", "HON", "IBM",
"INTC", "JNJ", "JPM", "MCD", "MRK",
"MSFT", "NKE", "PG", "CRM", "TRV",
"UNH", "VZ", "V", "WBA", "WMT")
Consider weekly (arithmetic) returns of these stocks from 2017-01-01 to 2022-1-31. We
use the data from 2017-01-01 to 2019-12-31 for training, and the data from 2020-01-01 to
2022-01-31 for testing. In R, the stock prices can be downloaded using the following codes:
library(quantmod)
start_date <- "2017-01-01"
end_date <- "2022-1-31"
R_all <- list()
for (j in 1:length(symbol_seq)) {
symbol <- symbol_seq[j]
price <- getSymbols(Symbols = c(symbol), src = "yahoo",
return.class = "xts",
from = start_date, to = end_date,
periodicity = "daily", auto.assign = FALSE)
R_all[[j]] <- weeklyReturn(price)
# check there are no missing values
cat("j = ", j, "\n", sep = "")
cat(sum(!is.na(R)), " ", sum(is.na(R)), "\n")
}
For the purposes of this problem, we assume that the weekly return vectors are i.i.d. over
time. Over the training period, we estimate the covariance matrix Σ using the empirical
covariance matrix.
(a) Consider the empirical covariance matrix Σ estimated from the training data. Us-
ing quadratic programming, find the global minimum variance portfolio under the
constraints
∑
iwi = 1 and wi ≥ 0 for all i.
(b) Find an equally weighted portfolio with 10 stocks such that the variance (of the
portfolio return) with respect to Σ is minimized. (Use the heuristics introduced in
the class.) Also find the 10 stocks whose variances are the smallest and form an
equally weighted portfolio.
(c) Consider the three portfolios found in (a) and (b). Compare the variances (of their
returns) in the testing period. Also plot density estimates of their return distributions.
Comment on your results.
3. (Taken from Chapter 3 of the book Optimization Methods in Finance (2nd edition), Cam-
bridge University Press) A company will face the following cash requirements in the next
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eight quarters (positive entries present cash needs while negative entries represent cash
surpluses):
Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8
100 500 100 ?600 ?500 200 600 ?900
The company has three borrowing possibilities:
A 2-year loan available at the beginning of Q1, with a 1% interest per quarter.
The other two borrowing opportunities are available at the beginning of every quarter:
a 6-month loan with a 1.8% interest per quarter, and a quarterly loan with a 2.5%
interest for the quarter.
Any surplus can be invested at a 0.5% interest per quarter.
Formulate a linear program that maximizes the wealth of the company at the beginning
of Q9. (You are invited to solve the problem using an available package, but the solution
is not required.)