Copyright 2011 - 2020 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/.

Last updated June 2018

```
x = 10; % assign x the value 10, silencing output print with ;
disp(x) % display x
```

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
```

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
```

Last updated July 2020

```
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!(Matrix{Float64}(undef, 2,3),NaN) # 2x3 Float64 matrix of NaNs - computationally better
v = fill(NaN, (2,3)) # 2x3 Float64 matrix of NaNs - direct
println(v) # Julia prints matrices in a single line
println(size(v)) # as expected, v is size (2,3)
w = repeat([1,2,3]', outer = [3,2]) # repeats matrix thrice by rows, twice 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
```

Last updated July 2020

```
y = [3.14; 15; 9.26; 5]; % List with semicolons is a column vector
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
```

Last updated July 2020

```
y = [3.14,15,9.26,5]
using Statistics; # load package needed
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
```

Last updated June 2018

```
mean(y) % mean
var(y) % variance
std(y) % unbiased standard deviation, by default
skewness(y) % skewness
kurtosis(y) % kurtosis
```

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)
```

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)
```

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
```

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
```

Last updated July 2020

```
## 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 July 2020)
## Supported dists: https://juliastats.org/Distributions.jl/stable/fit/
```

Last updated July 2020

```
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 - Needs Econometrics Toolbox
```

Last updated July 2020

```
using Random, HypothesisTests; # loading required packages
Random.seed!(100) # set random seed
x = rand(TDist(5), 500) # create hypothetical dataset x
println(JarqueBeraTest(x)) # Jarque-Bera test for normality
println(LjungBoxTest(x,20)) # Ljung-Box test for serial correlation
```

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
```

Last updated July 2020

```
using Plots;
Random.seed!(100)
x = rand(TDist(5), 60) # Create hypothetical dataset x
acf = autocor(x, 1:20) # autocorrelation for lags 1:20
pacf = autocor(x, 1:20) # partial autocorrelation for lags 1:20
## Plotting using Plots.jl
plot(bar(acf, title = "Autocorrelation", legend = false), bar(pacf, title = "Partial autocorrelation", legend = false))
```

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
```

Last updated July 2020

```
## For loops
for i in range(3,length = 5) # using range with the "length" option
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)
using Statistics;
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
using Random, Distributions;
Random.seed!(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
```

Last updated July 2020

```
y = normrnd(0, 1, 50, 1);
z = trnd(4, 50, 1);
subplot(2,2,1)
bar(y) % bar plot
title("Bar plot")
subplot(2,2,2)
plot(y) % line plot
title("Line plot")
subplot(2,2,3)
histogram(y) % histogram
title("Histogram")
subplot(2,2,4)
scatter(y,z) % scatter plot
title("Scatter plot")
```

Last updated July 2020

```
## For the simple plots in FRF we use Plots.jl
## 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)
## Plotting barplot, lineplot, histogram, scatterplot of y
return plot(bar(y, title = "Bar plot"), plot(y, title = "Line plot"),
histogram(y, title = "Histogram"), scatter(y, title = "Scatter plot"), legend = false)
## Wrapping plot(...) around multiple plots allows for automatic subplotting
## Options in wrapped plot(...) apply to all subplots
## Plot individual graphs using histogram(y), bar(y), etc. directly
## More examples using GR (plus syntax for customizations) can be found online:
## https://docs.juliaplots.org/latest/generated/gr/
```

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)
```

Last updated July 2020

```
## 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 = fit_mle(Normal, x) will return an object res of type Distribution
## with fitted parameters res.μ and res.σ
y = rand(Normal(0,1), 100)
res = fit_mle(Normal, y)
println("Fitted mean: ", res.μ)
println("Fitted sd: ", res.σ)
```