关于算法:ECMT3150-R语言统计汇总

ECMT3150: Assignment 1 (Semester 1, 2023)

1. [Total: 24 marks]
   Note: Please append your R codes (as a separate .R le) for part (g) while you submit the
   assignment.
   Let Xi denote the log-price of a stock, Cherry Inc. (code: CRRY), by the end of trading day
   i, and let Xi := Xi Xi1; thus Xi is the log-return on trading day i (i.e., over period
   (i 1; i]).
   Assume fXig_i0 follows the AR(1) model:
   Xi = 0 + 1Xi1 + ui: (1)
   where ui iid normal with mean 0 and variance 2.
   Let fFigi0 be the natural ?ltration generated by fuigi0.
   (a) [2 marks] Express Xi in terms of Xi1 and ui.
   (b) [2 marks] Compute E(XijFi1).
   (c) [2 marks] Compute V ar(XijFi1).
   (d) [2 marks] What is the condition on 0 and 1 such that fXigi1 is a martingale
   di¤erence sequence?
   A trading strategy is de?ned by figi0, where i is measurable with respect to Fi. Speci?-
   cally, i represents the number of CRRY shares a trader buys at the start of day i.
   The log-return due to the trading strategy over period (0; T] is given by
   rT =
   TX
   i=1
   i1Xi.
   (e) [4 marks] Alice invested in a share of CRRY using a buy-and-hold strategy, with i 1
   for all i. Compute E(rT) and V ar(rT) with 0 = 0 and 1 = 1.
   (f) [4 marks] Bob suggested another strategy, with i Xi for i > 0 and Compute E(rT)
   and V ar(rT) with 0 = 0 and 1 = 1.
   1
   (g) [8 marks] Carol suggested yet another strategy, with i 1fXi > 0g and 0 = 1.
   We want to evaluate the risk-return tradeo¤ of the proposed strategies using computer
   simulation.
   Start an R session, and set a random seed equal to the last 3 digits of your student ID.1
   Then generate B sample values of rT (name them as r
   (1)
   T ; r
   (2)
   T ; : : : ; r
   (B)
   T ), and compute
   the sample mean and variance of rT as follows:
   rT =
   1
   B
   BX
   b=1
   r
   (b)
   T ;
   se(rT) =
   1
   B 1
   BX
   b=1
   (r
   (b)
   T rT )2:
   For the purpose of your simulations, set T = 63, 2 = 0:1, B = 1000.
   The Sharpe ratio, de?ned as SR = rTse(rT) , is a common measure of the risk-return
   tradeo¤. Trading strategies with higher SR are more preferred by investors.
   Complete the following table with SR values. Comment on the performance of the
   trading strategies under di¤erent scenarios.
2. 1 Alice Bob Carol
3. 1
   0:01 1
   0:01 1
4. 0:9
5. 1:1
6. [Total: 16 marks] LetM denote the mood of Mimi (h: happy; a: angry), and let W denote
   the weather (s: sunny; r: rainy). The joint probability distribution of M and W is given in
   the table below. The row and column sums are displayed in the last column and in the last
   row, respectively.
   p(m;w) M = h M = a
   W = s 0:4 0:1
   W = r 0:2 0:3
   (a) [2 marks] Compute P (M = a).
   (b) [2 marks] Derive the conditional distribution of W given M = a.
   Assume that, given m and w, your test score S follows a normal distribution with mean
   (m;w) := E(SjM = m;W = w) and standard deviation 5. The conditional mean function
   (m;w) is given in the table below:
   1This is to ensure that your answers are replicable but di¤erent from those of other students.
   2
   (m;w) m = h m = a
   w = s 80 50
   w = r 70 40
   The passing score is 50 or above.
   (c) [3 marks] Compute the mean score E(S).
   (d) [3 marks] Given that Mimi was angry, what is the mean score you would get?
   (e) [3 marks] Compute the probability of failing the test.
   (f) [3 marks] Given that you failed the test, what is the probability that Mimi was angry?
7. [Total: 20 marks]
   Note: Please append your R codes (as a separate .R ?le) while you submit the assignment.
   Carol, an amateur economist, proposes the following time series model for unemployment
   rate:
   yt =
   1
   20
   +
   p
   3
   2
   yt1 1
   4
   yt2 + “t; (2)
   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 ?rst 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 y1 to be y0 = 0:1 and y1 = 0:12. While simulating the random
   numbers for “t, set the random seed to be your last 3 digits of your student ID.
   (e) [2 marks] Plot the sample ACF and record its value for the ?rst 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 ?intercept?estimate in the arima output is in fact the unconditional
   mean; see Rob Hyndman?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 con?dence interval for each period over the next 5 years.
   Describe the short-run and long-run behaviour of the point forecast and the con?dence
   interval.