代做EMET3007/6012 Business and Economic Forecasting Assignment 1代做留学生Matlab程序

Business and Economic Forecasting

EMET3007/6012

Assignment 1

•  100 % = 28 points.

• Maximal 3 students can submit one solution of Assignment 1. All stu- dents within a group receive the same grade. Both students of each group submit the SAME solution WITH uni-id’s of students within the group. Solutions are submitted via Wattle(Turniton). The other student(s) sub- mits his/her uni id and the uni id of students within the group.

• Name the uploaded file, if possible as a pdf, in the form of “your uni id-A1” .

• Only working within each group is allowed.  Please use wattle to build groups. Try to be precise in your argumentation.

• If there is an issue with uploading the file on wattle please send a copy to [email protected] and explain the nature of the technical issue.

• Due on 21st August 2024 WEDNESDAY 6pm (Week 5) (GMT+10). Late assignment will not be accepted.

Form. of Solution

• coding in matlab is expected; you can use parts of the code which is avail- able on wattle) however excel is feasible.

• please provide code (with explanation) in your solution file so that it can be easily replicated with matlab.

• please type your solutions

Solve the following 5 problems.

Problem 1 1+2+2=5 points

Suppose X1, . . . , Xn  are iid random variables with mean µ and variance σ2  < ∞ .

(a)  Find the expected value and variance of the sample mean

(b)  Show that

(c)  Show that the sample variance

is an unbiased estimator of σ2, i.e., E[S2] = σ2 .

Problem 2                                                           4 points

Write a MATLAB program to generate data from the trend-cycle model in Ex- ample 3 from Week 2 with a quadratic trend component.

mt  = a0 + a1t + a2t2,

and the same cycle component

ct  = b1cos(ωt) + b2 sin(ωt).

Plot time path of the model

yt  = mt + ct

with T = 50, a0  = 0, a1  = 0.5, a2  = 0.1, b1  = 4, b2  = 0, ω = 2π/4 and σ2  = 2.

Note: here is no noise component!

Problem 3 5+1=6 points

a. Write a MATLAB program to generate data from the following mean-reverting process:

yt  = µ + φ(yt -1 - µ) + et, et  ~ N(0, σ2 ),

fort = 1, 2, . . . , T, where the process is initialized with y0  = μ .  Plot one real- ization of the process with T = 50, μ = 5, φ = 0.8 and σ2  = 2.

b.   What can you say about the parameter φ and how is it affecting the process?

Note: here is a noise component!

Problem 4                                    3+3+3+2+1=12 points

Suppose we wish to compare the forecast accuracy of two simple methods for forecasting the Canberra inflation rate: (1) the mean using data up to the most recent observation, and (2) a random-walk method that uses only the most recent observation.  The evaluation period is from 2016 Q3 to 2021 Q4.  For

Table 1: Canberra CPI Inflation rate per quarter from ABS

Quarter	Inflation	Quarter	Inflation
2014 Q3	0.4	2018 Q2	0.4
2014 Q4	0.1	2018 Q3	0.6
2015 Q1	-0.1	2018 Q4	0.7
2015 Q2	0.4	2019 Q1	0.1
2015 Q3	0.2	2019 Q2	0.3
2015 Q4	0.2	2019 Q3	0.7
2016 Q1	0.2	2019 Q4	0.6
2016 Q2	0.2	2020 Q1	0.4
2016 Q3	0.8	2020 Q2	-2.3
2016 Q4	0.6	2020 Q3	2.3
2017 Q1	0.6	2020 Q4	0.8
2017 Q2	0.0	2021 Q1	0.9
2017 Q3	0.9	2021 Q2	0.8
2017 Q4	0.6	2021 Q3	1.3
2018 Q1	0.8	2021 Q4	1.0

measures of accuracy, we consider MAFE and MSFE.

a)  Use the two methods (historical mean and random walk) to produce one- step-ahead forecasts for each quarter from 2016 Q3 to 2021 Q4. For each method, graph your forecast against the actual data.

b)  Compute the two measures of forecast accuracy for the two methods. Which model performs better according to these measures?

c)  (can be solved without a) and b))

Find one additional measure of forecast accuracy (online or elsewhere). Describe this measure, and compare the two models under this new mea- sure.

d)  (can be solved without a) and b))

A large portion of the error in your forecasts is coming from a small num- ber of outlier datapoints.  Identify these datapoints, and describe what real-world event is driving this anomaly.

e)  (can be solved without a) and b))

Find the actual 2022 Q1 Canberra CPI inflation rate from the ABS. Com- pare the realisation to your forecast.

Problem 5 4 points

Prove the following theorem:  Given a density forecast f (yT+hjIT, θ) and the absolute loss function L(y(ˆ), y)  =  j y(ˆ) - yj, the point forecast which minimises expected loss is the median mT+h.

Note: the median is the unique value mT+h  such that y T+h is as equally likely to be above mT+h as below it. Mathematically, mT+his the unique value which satisfies

Clarification: The task in this question is to show that to minimise absolute

loss,the modeller should sety(ˆ)T+h  = mT+h.