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.

Last updated June 2018

```
x = 10 # assign x the value 10
print(x) # print the value of 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
```

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```
y = [1,3,5,7,9] # lists in square brackets are stored as arrays
print(y)
print(y[2]) # 3rd element (Python indices start at 0)
print(len(y)) # as expected, y has length 5
import numpy as np
v = np.full([2,3], np.nan) # create a 2x3 matrix with NaN values
print(v)
print(v.shape) # as expected, v is size (2,3)
w=np.tile(np.transpose([1,2,3]),(3,2)) # repeats twice by rows, thrice by columns
print(w)
s = range(10) # an iterator from 0 to 9
print([x for x in s]) # return elements using list comprehension
```

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
```

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```
## 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 numpy.loadtxt() to read it
##
## Example:
## using numpy as np
## data = np.loadtxt('data.csv', delimiter = ',', skiprows = 1)
## skiprows=1 ensures that the header row is skipped
```

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```
## 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
```

Last updated June 2018

```
import numpy as np
y = [3.14,15,9.26,5]
print(sum(y)) # sum of all elements of y
print(max(y)) # maximum value of y
print(min(y)) # minimum value of y
print(np.mean(y)) # arithmetic mean
print(np.median(y)) # median
print(np.var(y)) # variance
print(np.cov(y)) # covar matrix = variance for single vector
print(np.corrcoef(y)) # corr matrix = [1] for single vector
print(np.sort(y)) # sort in ascending order
print(np.log(y)) # natural log
```

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
```

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```
import numpy as np
from scipy import stats
print(np.mean(y)) # mean
print(np.var(y)) # variance
print(np.std(y, ddof = 1)) # ddof = 1 for unbiased standard deviation
print(stats.skew(y)) # skewness
print(stats.kurtosis(y, fisher = False)) # fisher = False gives Pearson definition
```

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```
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)
```

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```
import numpy as np
z = np.matrix([[1, 2], [3, 4]]) # z is a 2 x 2 matrix
x = np.matrix([1, 2]) # x is a 1 x 2 matrix
## Note: z * x is undefined since the two matrices are not conformable
print(z * np.transpose(x)) # this evaluates to a 2 x 1 matrix
b = np.concatenate((z,x), axis = 0) # "stacking" z and x vertically
print(b)
c = np.concatenate((z,np.transpose(x)), axis = 1) # "stacking" z and x horizontally
print(c)
## note: dimensions must match along the combining axis
```

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```
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
```

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```
import numpy as np
from scipy import stats
q = np.arange(-3,4,1) # specify a set of values
p = np.arange(0.1,1.0,0.1) # specify a set of probabilities
print(stats.norm.ppf(p)) # element-wise inverse Normal quantile
print(stats.t.cdf(q,4)) # element-wise cdf under Student-t(4)
print(stats.chi2.pdf(q,2)) # element-wise pdf under Chisq(2)
## One can also obtain pseudorandom samples from distributions using numpy.random
x = np.random.standard_t(df=5, size=100) # Sampling 100 times from TDist with 5 df
y = np.random.normal(size=50) # Sampling 50 times from a standard normal
## Given data, we obtain MLE estimates of parameters with stats:
res = stats.norm.fit(x) # Fitting x to normal dist
print(res)
```

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```
## 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
```

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```
from scipy import stats
from statsmodels.stats.diagnostic import acorr_ljungbox
x = np.random.standard_t(df=5, size=500) # Create dataset x
print(stats.jarque_bera(x)) # Jarque-Bera test
print(acorr_ljungbox(x, lags=20)) # Ljung-Box test
```

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
```

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```
import statsmodels.api as sm
import matplotlib.pyplot as plt
x = np.random.standard_t(df = 5, size = 60) # Create hypothetical dataset x
q1 = sm.tsa.stattools.acf(y, nlags=20) # autocorrelation for lags 1:20
plt.bar(x = np.arange(1,len(q1)), height = q1[1:])
plt.show()
plt.close()
q2 = sm.tsa.stattools.pacf(y, nlags=20) # partial autocorr for lags 1:20
plt.bar(x = np.arange(1,len(q2)), height = q2[1:])
plt.show()
plt.close()
```

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
```

Last updated June 2018

```
import numpy as np
## For loops
for i in range(3,8): # NOTE: range(start, end), end excluded
print(i**2) # range(3,8) iterates through [3,4,5,6,7)
## 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)
def excess_kurtosis(x, excess = 3): # note: excess optional, default = 3
m4=np.mean((x-np.mean(x))**4) # note: exponentiation in Python uses **
excess_kurt=m4/(np.std(x)**4)-excess
return excess_kurt
x = np.random.standard_t(df=5,size=60) # Create hypothetical dataset x
print(excess_kurtosis(x))
```

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
```

Last updated June 2018

```
import numpy as np
import matplotlib.pyplot as plt
y = np.random.normal(size = 50)
z = np.random.standard_t(df = 4, size = 50)
## using Matplotlib to plot bar, line, histogram and scatter plots
plt.subplot(2,2,1)
plt.bar(range(len(y)), y)
plt.subplot(2,2,2)
plt.plot(y)
plt.subplot(2,2,3)
plt.hist(y)
plt.subplot(2,2,4)
plt.scatter(y,z)
```

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/
```

Last updated June 2018

```
## Convert objects from one type to another with int(), float() etc
## To check type, use type(object)
x = 8.0
print(type(x))
x = int(x)
print(type(x))
```

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
```