Appendix - Introduction (in MATLAB/Julia)


Copyright 2011, 2016, 2018 Jon Danielsson. This code is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. This code is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. The GNU General Public License is available at: https://www.gnu.org/licenses/.
The original 2011 R code will not fully work on a recent R because there have been some changes to libraries. The latest version of the Matlab code only uses functions from Matlab toolboxes.
The GARCH functionality in the econometric toolbox in Matlab is trying to be too clever, but can't deliver and could well be buggy. If you want to try that, here are the docs (estimate). Besides, it can only do univariate GARCH and so can't be used in Chapter 3. Kevin Sheppard's MFE toolbox is much better, while not as user friendly, it is much better written and is certainly more comprehensive. It can be downloaded here and the documentation here is quite detailed.


Listing M.1: Entering and Printing Data
Last updated June 2018

x = 10; % assign x the value 10, silencing output print with ;
disp(x) % display x
		
Listing J.1: Entering and Printing Data in Julia
Last updated June 2018

x = 10     # assign x the value 10
println(x) # print x
## println() puts next output on new line, while print() doesn't
		

Listing M.2: Vectors, Matrices and Sequences
Last updated June 2018

y = [1,3,5,7,9]            % lists are denoted by square brackets
y(3)                       % calling 3rd element (MATLAB indices start at 1)
size(y)                    % shows that y is 1 x 5 (a row vector, by default)
length(y)                  % as expected, y has length 5
v = nan(2,3)               % fill a 2 x 3 matrix with NaN values
size(v)                    % as expected, v is size (2,3)
w = repmat([1,2,3]', 2, 3) % repeats matrix twice by rows, thrice by columns
s = 1:10                   % s is a list of integers from 1 to 10 inclusive
		
Listing J.2: Vectors, Matrices and Sequences in Julia
Last updated June 2018

y = [1,3,5,7,9]                    # lists in square brackets are stored as arrays
println(y)
println(y[3])                      # calling 3rd element (Julia indices start at 1)
println(size(y))                   # size of y
println(length(y))                 # as expected, y has length 5
v = fill!(Array{Float64}(2,3),NaN) # 2x3 Float64 matrix of NaNs
println(v)                         # Julia prints matrices in a single line
println(size(v))                   # as expected, v is size (2,3)
w = repmat([1,2,3], 2, 3)          # repeats matrix twice by rows, thrice by columns
println(w)
s = 1:10                           # s is an sequence which one can loop across
println(collect(s))                # return sequence elements as an array
		

Listing M.3: Importing Data
Last updated June 2018

%% There are many data sources for financial data, for instance
%% Yahoo Finance, AlphaVantage and Quandl. However, some of the
%% free data sources have numerous issues with accuracy and
%% handling of missing data, so only CSV importing is shown here.
%%
%% For csv data, one can use csvread to read it
%%
%% Example:
%% data = csvread('data.csv', 1, 0);
%% the two numbers behind are the row offset and column offset
%% so here we ignore the first row (ie. the header)
		
Listing J.3: Importing Data in Julia
Last updated June 2018

## There are many data sources for financial data, for instance
## Yahoo Finance, AlphaVantage and Quandl. However, some of the
## free data sources have numerous issues with accuracy and
## handling of missing data, so only CSV importing is shown here.
##
## For csv data, one can use the package CSV to read it
##
## Example:
## using CSV;
## data = CSV.read("data.csv", nullable = false)
## nullable = false avoids type problems involving NullableArray types
		

Listing M.4: Basic Summary Statistics
Last updated June 2018

y = [3.14,15,9.26,5];
sum(y)                % sum of all elements of y
prod(y)               % product of all elements of y
max(y)                % maximum value of y
min(y)                % minimum value of y
range(y)              % min, max value of y
mean(y)               % arithmetic mean
median(y)             % median
var(y)                % variance
cov(y)                % covar matrix = variance for single vector
corrcoef(y)           % corr matrix = [1] for single vector
sort(y)               % sorting in ascending order
log(y)                % natural log
		
Listing J.4: Basic Summary Statistics in Julia
Last updated June 2018

y = [3.14,15,9.26,5]
println("sum: ", sum(y))        # return sum of all elements of y
println("product: ", prod(y))   # return product of all elements of y
println("max: ", maximum(y))    # return maximum value of y
println("min: ", minimum(y))    # return minimum value of y
println("mean: ", mean(y))      # arithmetic mean
println("median: ", median(y))  # median
println("variance: ", var(y))   # variance
println("cov_matrix: ", cov(y)) # covar matrix = variance for single vector
println("cor_matrix: ", cor(y)) # corr matrix = [1] for single vector
println(sort(y))                # sorts y in ascending order
println(log.(y))                # natural log, note . denotes elementwise operation
		

Listing M.5: Calculating Moments
Last updated June 2018

mean(y)     % mean
var(y)      % variance
std(y)      % unbiased standard deviation, by default
skewness(y) % skewness
kurtosis(y) % kurtosis
		
Listing J.5: Calculating Moments in Julia
Last updated June 2018

using StatsBase;
println("mean: ", mean(y))         # mean
println("variance: ", var(y))      # variance
println("std dev: ", std(y))       # unbiased standard deviation
println("skewness: ", skewness(y)) # skewness
println("kurtosis: ", kurtosis(y)) # EXCESS kurtosis (note the different default)
		

Listing M.6: Basic Matrix Operations
Last updated June 2018

z = [1, 2; 3, 4] % z is a 2 x 2 matrix (Note the use of ; as row separator)
x = [1, 2]       % x is a 1 x 2 matrix
%% Note: z * x is undefined since the two matrices are not conformable
z * x'           % this evaluates to a 2 x 1 matrix
vertcat(z,x)     % "stacking" z and x vertically
horzcat(z,x')    % "stacking z and x' horizontally
%% Note: dimensions must match along the combining axis)
		
Listing J.6: Basic Matrix Operations in Julia
Last updated June 2018

z = Matrix([[1 2];[3 4]]) # z is a 2 x 2 matrix
x = Matrix([1 2])         # x is a 1 x 2 matrix
## Note: z * x is undefined since the two matrices are not conformable
println(z * x')           # this evaluates to a 2 x 1 matrix
b = vcat(z,x)             # "stacking" z and x vertically
c = hcat(z,x')            # "stacking" z and x' horizontally
## Note: dimensions must match along the combining axis
		

Listing M.7: Statistical Distributions
Last updated June 2018

q = -3:1:3                 % specify a set of values
p = 0.1:0.1:0.9            % specify a set of probabilities
norminv(p, 0, 1)           % element-wise inverse Normal quantile
tcdf(q, 4)                 % element-wise cdf under Student-t(4)
chi2pdf(q, 2)              % element-wise pdf under Chisq(2)
%% One can also obtain pseudorandom samples from distributions
x = trnd(5, 100, 1);       % Sampling 100 times from t dist with 5 df
y = normrnd(0, 1, 100, 1); % Sampling 50 times from a standard normal
%% Given sample data, we can also obtain MLE estimates of distribution parameters:
res = fitdist(x, "Normal") % Fitting x to normal dist
		
Listing J.7: Statistical Distributions in Julia
Last updated June 2018

## Julia has a wide range of functions contained in the package Distributions.jl
## Vectorized versions of the functions are used here as they are relevant for FRF
using Distributions;
q = collect((-3:1:3))             # specify a set of values
p = collect((0.1:0.1:0.9))        # specify a set of probabilities
println(quantile.(Normal(0,1),p)) # element-wise inverse Normal quantile
println(cdf.(TDist(4), q))        # element-wise cdf calculation under Student-t(4)
println(pdf.(Chisq(2), q))        # element-wise pdf calculation under Chisq(2)
## Similar syntax for other dists, e.g. Bernoulli(p), Binomial(n,p), Poisson(λ)
## For full list of supported distributions, see Distributions.jl documentation
## One can also obtain pseudorandom samples from distributions using rand()
x = rand(TDist(5), 100)           # Sampling 100 times from TDist with 5 df
y = rand(Normal(0,1), 50)         # Sampling 50 times from a standard normal
## Given data, we obtain MLE estimates of parameters with fit_mle():
fit_mle(Normal, x)                # Fitting x to normal dist
## Some distributions like the Student-t cannot be fitted yet (as of June 2018)
## Supported dists: https://juliastats.github.io/Distributions.jl/latest/fit.html#Applicable-distributions-1
		

Listing M.8: Statistical Tests
Last updated June 2018

x = trnd(5, 500, 1);                    % Create hypothetical dataset x
[h1, p1, jbstat] = jbtest(x)            % Jarque-Bera test for normality
[h2, p2, lbstat] = lbqtest(x,'lags',20) % Ljung-Box test for serial correlation
		
Listing J.8: Statistical Tests in Julia
Last updated June 2018

srand(100)
x = rand(TDist(5), 500)     # Create hypothetical dataset x
## We use the package HypothesisTests
using HypothesisTests;
println(JarqueBeraTest(x))  # Jarque-Bera test for normality
println(LjungBoxTest(x,20)) # Ljung-Box test for serial correlation
		

Listing M.9: Time Series
Last updated June 2018

x = trnd(5, 60, 1); % Create hypothetical dataset x
subplot(1,2,1)
autocorr(x, 20)     % autocorrelation for lags 1:20
subplot(1,2,2)
parcorr(x,20)       % partial autocorrelation for lags 1:20
		
Listing J.9: Time Series in Julia
Last updated June 2018

srand(100)
x = rand(TDist(5), 60)    # Create hypothetical dataset x
using Plots, StatsBase;   # refer to Listing 0.11 for Plots.jl
acf = autocor(x, 1:20)    # autocorrelation for lags 1:20
pacf = autocor(x, 1:20)   # partial autocorrelation for lags 1:20
plot(bar(acf), bar(pacf)) # plotting the ACF/PACF using Plots.jl
		

Listing M.10: Loops and Functions
Last updated June 2018

%% For loops
for i = 3:7                          % iterates through [3,4,5,6,7]
    i^2
end
%% If-else loops
X = 10;
if (rem(X,3) == 0)
    disp("X is a multiple of 3")
else
    disp("X is not a multiple of 3")
end
%% Functions (example: a simple excess kurtosis function)
%% NOTE: in MATLAB, functions can be defined in 2 locations:
%% 1) in a separate file (e.g. excess_kurtosis.m in this case) in the workspace
%% 2) in the same file as the rest of the code, BUT at the end of the file
%% function k = excess_kurtosis(x, excess)
%%     if nargin == 1                % if there is only 1 argument
%%         excess = 3;               % set excess = 3
%%     end                           % this is how optional param excess is set
%%     m4 = mean((x-mean(x)).^4);
%%     k = m4/(std(x)^4) - excess;
%% end
		
Listing J.10: Loops and Functions in Julia
Last updated June 2018

## We demonstrate how loops and functions work in Julia with some examples
## Main differences from Python
## 1) No semicolons on the first line of loops/functions
## 2) insert "end" after the last line of loops/functions
## 3) Note: difference in range(.) function between Python and Julia (see below)
## For loops
for i in range(3,5)                              # NOTE: range(start,n) unusual!
    println(i^2)                                 # where n = number of terms
    end                                          # this iterates over [3,4,5,6,7]
## If-else loops
X = 10
if X % 3 == 0
    println("X is a multiple of 3")
else
    println("X is not a multiple of 3")
end
## Functions (example: a simple excess kurtosis function)
function excess_kurtosis(x, excess = 3)::Float64 # excess optional, default = 3
    m4 = mean((x-mean(x)).^4)                    # element-wise exponentiation .^
    excess_kurt = m4/(std(x)^4) - excess
    return excess_kurt
end
srand(100)
x = rand(TDist(5), 60)                           # Create hypothetical dataset x
excess_kurtosis(x)
## Note: we have forced output to be of type Float64 by the type declaration above
		

Listing M.11: Basic Graphs
Last updated June 2018

y = normrnd(0, 1, 50, 1);
z = trnd(4, 50, 1);
subplot(2,2,1)
bar(y)                    % bar plot
subplot(2,2,2)
plot(y)                   % line plot
subplot(2,2,3)
histogram(y)              % histogram
subplot(2,2,4)
scatter(y,z)              % scatter plot
		
Listing J.11: Basic Graphs in Julia
Last updated June 2018

## For the simple plots in FRF we use Plots.jl for plotting
## Full documentation at http://docs.juliaplots.org/latest/
## By default, Plots.jl uses the GR backend, sufficient for plots done in FRF
## Alternative backends are also available, e.g. Plotly, PlotlyJS
y = rand(Normal(0,1), 50)
using Plots;
## plot barplot, lineplot, histogram, scatterplot of y
return plot(bar(y), plot(y), histogram(y), scatter(y))
## Wrapping plot(...) around multiple plots allows for automatic subplotting
## This can, of course, be manually specified too
## Plot individual graphs using histogram(y), bar(y) etc. directly
## More examples using GR (plus syntax for customizations) can be found online:
## http://docs.juliaplots.org/latest/examples/gr/
		

Listing M.12: Miscellaneous Useful Functions
Last updated June 2018

%% Convert objects from one type to another with int8() etc
%% To check type, use isfloat(object), isinteger(object) and so on
x = 8.0;
isfloat(x)
x = int8(x);
isinteger(x)
		
Listing J.12: Miscellaneous Useful Functions in Julia
Last updated June 2018

## 1) To convert objects from one type to another, use convert(Type, object)
##    To check type, use typeof(object)
x = 8.0
println(typeof(x))
x = convert(Int, 8.0)
println(typeof(x))
## 2) To type Greek letters, type \ + name + Tab in succession
##    e.g. \gammaTab gives you γ and \GammaTab gives you Γ
##
##    Greek letters are sometimes essential in retrieving parameters from functions
##    e.g. res = mle_fit(Normal, x) will return an object res of type Distribution
##    with fitted parameters res.μ and res.σ