ECMT3150: Assignment 1 (Semester 1, 2022)
Due: 5pm, 18 March 2022 (Friday)
1. [Total: 20 marks] Bob is a grown-up now. With the pocket money from his dad Simon, Bob went to the casino to try his luck. Being a Örst-time gambler, he decided to start by betting big-and-small.
For the sake of this question, we assume the game is fair in the sense that all rounds are independent and the probabilities of winning and losing in a round are both equal to 0.5. More precisely, we deÖne the indicator yn as follows:
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yn =( 1 ifBobwinsinthenthround, 1 if Bob loses in the nth round.
The assumptions on the game given above imply that (yn)n1 is a sequence of iid random variables with P(yn = 1) = P(yn = 1) = 0:5.
Let Vn (in dollars) denote Bobís stake in the nth round. The stake is determined by Bob before the nth round starts. If Bob bets Vn in the nth round and wins, he will gain Vn in that round (hence receiving a total of 2Vn); otherwise he will lose Vn (hence receiving a total of 0 dollars). Note that Vn may depend on the outcomes of the Örst n 1 rounds. A given sequence (Vn)n1 deÖnes a betting strategy.
Let Xn denote Bobís total gain after the nth round. We set X0 = 0. It is clear that Xn is
(a) [4 marks] DeÖne Xn := Xn Xn 1 for n 1. Show that the sequence (Xn)n1 is a
martingale di§erence sequence.
(b) [3 marks] Show that E(Xn) = 0 for all n. In words, Bob will break even on average.
Suppose Bob adopts the ìgrandpaís strategyîby betting $1 in each round regardless of what happened, i.e., Vn 1 for all n.
(c) [3 marks] What is the distribution of Bobís total gain after 5 rounds?
(d) [2 marks] What is the probability that Bob will su§er a loss after 5 rounds?
Bob claims that he can beat the house by adopting the following betting strategy (letís call it ìthe bankerís strategyî): He bets $1 in the Örst round. If Bob wins in the previous round, he stops playing; otherwise he doubles the bet in the following round. Bob will keep playing until he wins for the Örst time or he uses up all the money that Simon gave him, whichever happens Örst. More precisely, we can represent the bankerís strategy as follows: V1 = 1, and forn>1,
( 2n 1 ify1 = 1,y2 = 1,…,yn 1 = 1, Vn = 0 otherwise.
(e) [2 marks] What is Bobís total loss after he loses all the Örst 10 rounds, supposing that Bob has enough pocket money to survive beyond 10 rounds?
(f) [2 marks] What is Bobís total gain after he wins for the Örst time?
(g) [4 marks] Bob claims that by adopting the bankerís strategy he will for sure bring back home with more money than what Simon gave him if he plays long enough. Assuming that the game is fair, do you agree with Bob? Why or why not?
2. [Total: 16 marks] Let It be the indicator for the stock price movement in period t (-1 = price decrease; 0 = no change in price; 1 = price increase). Suppose the price movement is a Markov process in the sense that the price movement in the current period depends only on the price movement in the previous period. The joint probability distribution of It 1 and It is given in the table below, where p00, p01, p10, and p11 are constants. The row and column sums are displayed in the last column and in the last row, respectively.
It 1 = 1 It 1 = 0 It 1 = 1
It= 1 It=0 It=1
0.04 0.14 0.14 p00 0.02 p10 0.2 0.7
p01 0.7 p11 0.1 0.1
(a) [2 marks] Find E(It) and V ar(It).
(b) [2 marks] Is the process (It)t1 stationary? Explain.
(c) [3 marks] Find the conditional distribution of It given that the price increases at time t 1. Express your answer in terms of p11.
(d) [3 marks] Find the conditional variance of It given that the price increases at time t 1. Express your answer in terms of p11.
(e) [2 marks] Suppose the stock price movement are independent. Find the value of p01.
(f) [4 marks] Suppose the stock price movement are positively correlated. Find the range of values of p00.
3. [Total: 20 marks]
Note: Please append your R codes (as a separate .R Öle) while you submit the assignment.
Carol, an amateur economist, proposes the following time series model for unemployment
yt= 1 +p3yt 1 1yt 2+”t; (1) 20 2 4
where “t iid N(0;0:022) (normal distribution with mean 0 and variance 0:022). The time period is measured in number of quarters.
(a) [3 marks] Show that the time series fytg generated by model (1) is stationary.
(b) [3 marks] There is a stochastic cycle in the time series generated by model (1). Find its
periodity in number of quarters.
(c) [4 marks] Compute the ACF for the Örst 3 lags, i.e., (1), (2) and (3).
(d) [2 marks] Write an R program to simulate a sample path of fytg over 30 years. Set the initial values y0 and y 1 to be y0 = 0:1 and y 1 = 0:12. While simulating the random numbers for “t, set the random seed to be your last 5 digits of your SID (this is to ensure that your answer will be di§erent from that of other students).
(e) [2 marks] Plot the sample ACF and record its value for the Örst 3 lags (the values can be retrieved from the acf command output stored as a list). Why are they di§erent from your answers in part (c)?
(f) [3 marks] Using the simulated sample path in part (d), estimate an AR(2) model using the R command arima. Write down the estimated model with the parameter estimates and their standard error. Also record the estimated variance of the innovations. [Important note: the ìinterceptîestimate in the arima output is in fact the unconditional mean; see ́s page for details: https://robjhyndman.com/hyndsight/ arimaconstants/.]
(g) [3 marks] Using the simulated sample path in part (d) and the R package forecast, plot the point forecast and the conÖdence interval for each period over the next 5 years. Describe the short-run and long-run behaviour of the point forecast and the conÖdence interval.
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