Create a function that implements the Generalized Method of Wavelet Moments for specific stochastic processes (or time series).

INSTRUCTIONS TO CANDIDATES

ANSWER ALL QUESTIONS

For this evaluation you will be asked to complete two consecutive tasks:

1.Create a function that implements the Generalized Method of Wavelet Moments (GMWM) for specific stochastic processes (or time series). The GMWM is defined as

θ :=argmin (ν − ν(θ))⊺ Ω (ν − ν(θ)),θ∈Θ 

where

θ is the p × 1 parameter vector of the time series model we want to estimate

ν is the J × 1 vector of Haar wavelet variance estimated on a time series, where J = ⌊log2(T )⌋

and T is the length of the time series

ν(θ) is J × 1 vector of the theoretical wavelet variance which is implied by the model of interest (i.e. it is a function of the model parameters)

Ω is a J × J weighting matrix (usually the identity matrix)

2.Program a simulation study (similar to the one presented in this section of an online book (https://smac- group.github.io/ts/the-family-of-autoregressive-moving-average-models.html#estimation-of-arp-models))

verifying how well the GMWM estimates the true parameter vector θ0 that will be provided to you. You will

produce boxplots and compute the Median Absolute Error (MAE) defined as

MAE := median(|θ − θ0|).

In order to complete these tasks you will make use of the wv package (which computes ν) and consult this link (https://smac-group.com/computing/2016/05/23/process-to-haar-wavelet-variance-formulae.html#) (to understand

the possible forms for ν(θ)) in which νj(⋅) represents the jth element of the vector and τj  := 2j . You will also

use the simts package (and the gen_gts() function) to simulate the time series (more info on this package and time series at this link (https://cran.r-project.org/web/packages/simts/vignettes/vignettes.html))

Grading

This evaluation will affect the lowest grade among your assignments. Let x represent your lowest grade, let y represent the grade you obtain for this evaluation and let z (a value between 0 and 5) represent the instructor’s evaluation of the student’s effort put into this evaluation. Then your lowest grade x will be replaced based on the following logic:

 if(y > x) { x <- y

}

else

{

x <- x + z

}

Attachments:

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