# Appendix - Introduction (in R/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 R.1: Entering and Printing Data in R Last updated June 2018

x = 10   # assign x the value 10
print(x) # print 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 R.2: Vectors, Matrices and Sequences in R Last updated June 2018

y = c(1,3,5,7,9)                   # create vector using c()
print(y)
print(y[3])                        # calling 3rd element (R indices start at 1)
print(dim(y))                      # gives NULL since y is a vector, not a matrix
print(length(y))                   # as expected, y has length 5
v = matrix(nrow=2,ncol=3)          # fill a 2 x 3 matrix with NaN values (default)
print(dim(v))                      # as expected, v is size (2,3)
w = matrix(c(1,2,3),nrow=6,ncol=3) # repeats matrix twice by rows, thrice by columns
print(w)
s = 1:10                           # s is a list of integers from 1 to 10 inclusive
print(s)

##### 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 R.3: Importing Data in R 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 read.csv to read it
##
## Example:
## one can use the zoo() function from the package zoo
## to turn the data into a timeseries (see Listing 1.1/1.2)

##### 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 R.4: Basic Summary Statistics in R Last updated June 2018

y=matrix(c(3.1,4.15,9))
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
cor(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 R.5: Calculating Moments in R Last updated June 2018

library(moments)
mean(y)          # mean
var(y)           # variance
sd(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 R.6: Basic Matrix Operations in R Last updated June 2018

z = matrix(c(1,2,3,4),2,2) # z is a 2 x 2 matrix
x = matrix(c(1,2),1,2)     # x is a 1 x 2 matrix
## Note: z * x is undefined since the two matrices are not conformable
z %*% t(x)                 # this evaluates to a 2 x 1 matrix
rbind(z,x)                 # "stacking" z and x vertically
cbind(z,t(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 R.7: Statistical Distributions in R Last updated June 2018

q = seq(from = -3, to = 3, length = 7)    # specify a set of values
p = seq(from = 0.1, to = 0.9, length = 9) # specify a set of probabilities
qnorm(p, mean = 0, sd = 1)                # element-wise inverse Normal quantile
pt(q, df = 4)                             # element-wise cdf under Student-t(4)
dchisq(q, df = 2)                         # element-wise pdf under Chisq(2)
## Similar syntax for other distributions
## q for quantile, p for cdf, d for pdf
## followed by the abbreviation of the distribution
## One can also obtain pseudorandom samples from distributions
x = rt(100, df = 5)                       # Sampling 100 times from TDist with 5 df
y = rnorm(50, mean = 0, sd = 1)           # Sampling 50 times from a standard normal
## Given data, we obtain MLE estimates of distribution parameters with package MASS:
library(MASS)
res = fitdistr(x, densfun = "normal")     # Fitting x to normal dist
print(res)

##### 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 R.8: Statistical Tests in R Last updated June 2018

library(tseries)
x = rt(500, df = 5)                          # Create hypothetical dataset x
jarque.bera.test(x)                          # Jarque-Bera test for normality
Box.test(x, lag = 20, type = c("Ljung-Box")) # 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 R.9: Time Series in R Last updated June 2018

x = rt(60, df = 5) # Create hypothetical dataset x
acf(x,20)          # autocorrelation for lags 1:20
pacf(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 R.10: Loops and Functions in R Last updated June 2018

## For loops
for (i in 3:7)                             # iterates through [3,4,5,6,7]
print(i^2)
## If-else loops
X = 10
if (X %% 3 == 0) {
print("X is a multiple of 3")
} else {
print("X is not a multiple of 3")
}
## Functions (example: a simple excess kurtosis function)
excess_kurtosis = function(x, excess = 3){ # note: excess optional, default=3
m4 = mean((x-mean(x))^4)
excess_kurt = m4/(sd(x)^4) - excess
excess_kurt
}
x = rt(60, df = 5)                         # Create hypothetical dataset x
excess_kurtosis(x)

##### 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 R.11: Basic Graphs in R Last updated June 2018

y = rnorm(50, mean = 0, sd = 1)
par(mfrow=c(2,2))               # sets up space for subplots
barplot(y)                      # bar plot
plot(y,type='l')                # line plot
hist(y)                         # histogram
plot(y)                         # 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 R.12: Miscellaneous Useful Functions in R Last updated June 2018

## Convert objects from one type to another with as.integer() etc
## To check type, use typeof(object)
x = 8.0
print(typeof(x))
x = as.integer(x)
print(typeof(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