You are provided with the VIX dataset, which is widely used in financial market research. The dataset contains 8,920 observations of daily VIX values (vix) from 1990 to 2025. It also

Description
The VIX Index, often called the “fear gauge”, measures the market’s expectations of near-term
volatility based on S&P 500 option prices. Predicting the VIX index is useful because it helps
investors anticipate market volatility and manage risk more effectively in their investment strate gies. In this assignment, you are conducting a study that compares the predictive performance
between four families of basis functions: piece-wise constant, piece-wise linear, radial, and Laplace,
for a linear basis function (LBF) model designed to predict the VIX index value. The aim is to
investigate which family of basis functions is most suited for modelling the relationship between
time and volatility (measured by VIX).
You are provided with the VIX dataset, which is widely used in financial market research. The
dataset contains 8,920 observations of daily VIX values (vix) from 1990 to 2025. It also contains代写BUSS6002 fear gauge
the year (year) for which the value is observed. A scatter plot of the dataset is shown in Figure 1.
Figure 1: VIX levels from 1990 to 2025.
The specific LBF model being considered in your study is given by
y = ϕ(x)
⊤β + ε,
where y is the VIX value, x is year, and ε is a random noise; ϕ(x) denotes the vector of basis
function values; the parameter vector to be estimated is β. Four families of basis functions are
considered for computing ϕ(x); the first family is the set of piece-wise constant basis functions
ϕ(x) := [1, γ1(x), . . . , γk(x)]⊤, with
γi(x) := I(x > ti),
where I(x > ti) is an indicator function defined by
I(x > ti) := ( 1 if x > ti
0 if x ≤ ti
.
The break points {ti}
k
i=1 are calculated according to
ti
:= xmin +
i(xmax − xmin)
k + 1
, (1)
2
where xmin and xmax denote the smallest and largest observed values of x, respectively. The second
family is the set of piece-wise linear basis functions ϕ(x) := [1, x, λ1(x), . . . , λk(x)]⊤, with
λi(x) := (x − ti)I(x > ti),
where ti
is given by Equation (1). The third family is the set of radial basis functions ϕ(x) :=
[1, ρ1(x), . . . , ρk(x)]⊤, with
ρi(x) := exp  −
(x −
8
ti)
2
,
where ti
is given by Equation (1). The final family is the set of Laplace basis functions ϕ(x) :=
[1, τ1(x), . . . , τk(x)]⊤, with
τi(x) := exp  −
|x −
8
ti
|

,
where ti
is given by Equation (1).
Before comparing the four basis function families, you must set the number of components k
for all models. This hyperparameter value for each basis function family should be selected using
a validation set, by minimising the validation mean squared error (MSE).
You should select the optimal values of k by exhaustively searching through an equally-spaced
grid from 1 to 30, with a spacing of 1:
K := {1, 2, . . . , 30}.
Once the optimal values of the hyperparameters are chosen for all basis function families, you will
be able to compare the predictive performance between the four using a test set (i.e., by comparing
the test MSE between the four optimally selected models).
3
Report Structure
Your report must contain the following four sections:
Report Title
1 Introduction (0.5 pages)
– Provide a brief project background so that the reader of your report can understand
the general problem that you are solving.
– Motivate your research question.
– State the aim of your project.
– Provide a short summary of each of the rest of the sections in your report (e.g., “The
report proceeds as follows: Section 2 presents . . . ”).
2 Methodology (2 pages)
– Define and describe the LBF model.
– Define and describe the four choices of basis function families being investigated.
– Describe how the parameter vector β is estimated given the value of the hyperparameter
k. Discuss any potential numerical issues associated with the estimation procedure.
– Describe how the hyperparameter value can be determined automatically from data (as
opposed to manually setting the hyperparameter to an arbitrary value).
– Describe how the performance of the four families of basis functions is compared given
the optimal hyperparameter value.
3 Empirical Study (2.5 pages)
– Describe the datasets used in your study.
– Present (in a table) the selected hyperparameter value for each basis function family.
– Describe and discuss the table of selected hyperparameters.
– Visually present (using plots) the predicted response values for each basis function
family in the test set.
– Describe and discuss the plots of predicted values.
– Present (in a table) the test MSE values for each basis function family.
– Describe and discuss the table of test MSE values.
– Report the VIX forecasts of 2026, 2027, and 2028, given by the model with the smallest
test MSE. Include a brief description of how these forecasts are obtained.
4 Conclusion (0.5 pages)
– Discuss your overall findings / insights.
– Discuss any limitations of your study.
– Suggest potential directions of extending your study.
4
Rubric
This assignment is worth 30% of the unit’s marks. The assessment is designed to test your compu tational skills in implementing algorithms and conducting empirical experiments, as well as your
communication skills in writing a concise and coherent report presenting your approach and results.
The mark allocation across assessment items is given in Table 1.
Assessment Item Goal Marks
Section 1 Introduction 4
Section 2 Methodology 10
Section 3 Empirical Study 16
Section 4 Conclusion 3
Overall Presentation Clear, concise, coherent, and correct 5
Jupyter Notebook Reproducable results 2
Total 40
Table 1: Assessment Items and Mark Allocation