Code should execute sequentially if run in a Jupyter notebook

- See the set up page to install Jupyter, Python and all necessary libraries
- Please direct feedback to contact@quantecon.org or the discourse forum

# An Introductory Example¶

Contents

We’re now ready to start learning the Python language itself

The level of this and the next few lectures will suit those with some basic knowledge of programming

But don’t give up if you have none—you are not excluded

You just need to cover a few of the fundamentals of programming before returning here

Good references for first time programmers include:

- The first 5 or 6 chapters of How to Think Like a Computer Scientist
- Automate the Boring Stuff with Python
- The start of Dive into Python 3

Note: These references offer help on installing Python but you should probably stick with the method on our set up page

You’ll then have an outstanding scientific computing environment (Anaconda) and be ready to move on to the rest of our course

## Overview¶

In this lecture we will write and then pick apart small Python programs

The objective is to introduce you to basic Python syntax and data structures

Deeper concepts will be covered in later lectures

## The Task: Plotting a White Noise Process¶

Suppose we want to simulate and plot the white noise process \(\epsilon_0, \epsilon_1, \ldots, \epsilon_T\), where each draw \(\epsilon_t\) is independent standard normal

In other words, we want to generate figures that look something like this:

We’ll do this several different ways

## Version 1¶

Here’s a few lines of code that perform the task we set

```
import numpy as np
import matplotlib.pyplot as plt
x = np.random.randn(100)
plt.plot(x)
plt.show()
```

Let’s break this program down and see how it works

### Import Statements¶

The first two lines of the program import functionality

The first line imports NumPy, a favorite Python package for tasks like

- working with arrays (vectors and matrices)
- common mathematical functions like cos and sqrt
- generating random numbers
- linear algebra, etc.

After import numpy as np we have access to these attributes via the syntax np.

Here’s another example

```
import numpy as np
np.sqrt(4)
```

```
2.0
```

We could also just write

```
import numpy
numpy.sqrt(4)
```

```
2.0
```

But the former method is convenient and more standard

#### Why all the imports?¶

Remember that Python is a general purpose language

The core language is quite small so it’s easy to learn and maintain

When you want to do something interesting with Python, you almost always need to import additional functionality

Scientific work in Python is no exception

Most of our programs start off with lines similar to the import statements seen above

#### Packages¶

As stated above, NumPy is a Python *package*

Packages are used by developers to organize a code library

In fact a package is just a directory containing

- files with Python code — called
**modules**in Python speak - possibly some compiled code that can be accessed by Python (e.g., functions compiled from C or FORTRAN code)
- a file called __init__.py that specifies what will be executed when we type import package_name

In fact you can find and explore the directory for NumPy on your computer easily enough if you look around

On this machine it’s located in

```
anaconda3/lib/python3.6/site-packages/numpy
```

### Importing Names Directly¶

Recall this code that we saw above

```
import numpy as np
np.sqrt(4)
```

```
2.0
```

Here’s another way to access NumPy’s square root function

```
from numpy import sqrt
sqrt(4)
```

```
2.0
```

This is also fine

The advantage is less typing if we use sqrt often in our code

The disadvantage is that, in a long program, these two lines might be separated by many other lines

Then it’s harder for readers to know where sqrt came from, should they wish to

## Alternative Versions¶

Let’s try writing some alternative versions of our first program

Our aim in doing this is to illustrate some more Python syntax and semantics

The programs below are less efficient but

- help us understand basic constructs like loops
- illustrate common data types like lists

### A Version with a For Loop¶

Here’s a version that illustrates loops and Python lists

```
ts_length = 100
ϵ_values = [] # Empty list
for i in range(ts_length):
e = np.random.randn()
ϵ_values.append(e)
plt.plot(ϵ_values)
plt.show()
```

In brief,

- The first pair of lines
`import`

functionality as before - The next line sets the desired length of the time series
- The next line creates an empty
*list*called`ϵ_values`

that will store the \(\epsilon_t\) values as we generate them - The next three lines are the for loop, which repeatedly draws a new random number \(\epsilon_t\) and appends it to the end of the list
`ϵ_values`

- The last two lines generate the plot and display it to the user

Let’s study some parts of this program in more detail

### Lists¶

Consider the statement `ϵ_values = []`

, which creates an empty list

Lists are a *native Python data structure* used to group a collection of objects

For example, try

```
x = [10, 'foo', False] # We can include heterogeneous data inside a list
type(x)
```

```
list
```

The first element of `x`

is an integer, the next is a string and the third is a Boolean value

When adding a value to a list, we can use the syntax `list_name.append(some_value)`

```
x
```

```
[10, 'foo', False]
```

```
x.append(2.5)
x
```

```
[10, 'foo', False, 2.5]
```

Here `append()`

is what’s called a *method*, which is a function “attached to” an object—in this case, the list `x`

We’ll learn all about methods later on, but just to give you some idea,

- Python objects such as lists, strings, etc. all have methods that are used to manipulate the data contained in the object
- String objects have string methods, list objects have list methods, etc.

Another useful list method is `pop()`

```
x
```

```
[10, 'foo', False, 2.5]
```

```
x.pop()
```

```
2.5
```

```
x
```

```
[10, 'foo', False]
```

The full set of list methods can be found here

Following C, C++, Java, etc., lists in Python are zero based

```
x
```

```
[10, 'foo', False]
```

```
x[0]
```

```
10
```

```
x[1]
```

```
'foo'
```

### The For Loop¶

Now let’s consider the `for`

loop from the program above, which was

```
for i in range(ts_length):
e = np.random.randn()
ϵ_values.append(e)
```

Python executes the two indented lines `ts_length`

times before moving on

These two lines are called a `code block`

, since they comprise the “block” of code that we are looping over

Unlike most other languages, Python knows the extent of the code block *only from indentation*

In our program, indentation decreases after line `ϵ_values.append(e)`

, telling Python that this line marks the lower limit of the code block

More on indentation below—for now let’s look at another example of a `for`

loop

```
animals = ['dog', 'cat', 'bird']
for animal in animals:
print("The plural of " + animal + " is " + animal + "s")
```

If you put this in a text file or Jupyter cell and run it you will see

```
The plural of dog is dogs
The plural of cat is cats
The plural of bird is birds
```

This example helps to clarify how the `for`

loop works: When we execute a
loop of the form

```
for variable_name in sequence:
<code block>
```

The Python interpreter performs the following:

- For each element of
`sequence`

, it “binds” the name`variable_name`

to that element and then executes the code block

The `sequence`

object can in fact be a very general object, as we’ll see
soon enough

### Code Blocks and Indentation¶

In discussing the `for`

loop, we explained that the code blocks being looped over are delimited by indentation

In fact, in Python **all** code blocks (i.e., those occurring inside loops, if clauses, function definitions, etc.) are delimited by indentation

Thus, unlike most other languages, whitespace in Python code affects the output of the program

Once you get used to it, this is a good thing: It

- forces clean, consistent indentation, improving readability
- removes clutter, such as the brackets or end statements used in other languages

On the other hand, it takes a bit of care to get right, so please remember:

The line before the start of a code block always ends in a colon

`for i in range(10):`

`if x > y:`

`while x < 100:`

- etc., etc.

All lines in a code block

**must have the same amount of indentation**The Python standard is 4 spaces, and that’s what you should use

#### Tabs vs Spaces¶

One small “gotcha” here is the mixing of tabs and spaces, which often leads to errors

(Important: Within text files, the internal representation of tabs and spaces is not the same)

You can use your `Tab`

key to insert 4 spaces, but you need to make sure it’s configured to do so

If you are using a Jupyter notebook you will have no problems here

Also, good text editors will allow you to configure the Tab key to insert spaces instead of tabs — trying searching on line

### While Loops¶

The `for`

loop is the most common technique for iteration in Python

But, for the purpose of illustration, let’s modify the program above to use a `while`

loop instead

```
ts_length = 100
ϵ_values = []
i = 0
while i < ts_length:
e = np.random.randn()
ϵ_values.append(e)
i = i + 1
plt.plot(ϵ_values)
plt.show()
```

Note that

- the code block for the
`while`

loop is again delimited only by indentation - the statement
`i = i + 1`

can be replaced by`i += 1`

### User-Defined Functions¶

Now let’s go back to the `for`

loop, but restructure our program to make the logic clearer

To this end, we will break our program into two parts:

A

*user-defined function*that generates a list of random variablesThe main part of the program that

- calls this function to get data
- plots the data

This is accomplished in the next program

```
def generate_data(n):
ϵ_values = []
for i in range(n):
e = np.random.randn()
ϵ_values.append(e)
return ϵ_values
data = generate_data(100)
plt.plot(data)
plt.show()
```

Let’s go over this carefully, in case you’re not familiar with functions and how they work

We have defined a function called `generate_data()`

as follows

`def`

is a Python keyword used to start function definitions`def generate_data(n):`

indicates that the function is called`generate_data`

, and that it has a single argument`n`

- The indented code is a code block called the
*function body*—in this case it creates an iid list of random draws using the same logic as before - The
`return`

keyword indicates that`ϵ_values`

is the object that should be returned to the calling code

This whole function definition is read by the Python interpreter and stored in memory

When the interpreter gets to the expression `generate_data(100)`

, it executes the function body with `n`

set equal to 100

The net result is that the name `data`

is *bound* to the list `ϵ_values`

returned by the function

### Conditions¶

Our function `generate_data()`

is rather limited

Let’s make it slightly more useful by giving it the ability to return either standard normals or uniform random variables on \((0, 1)\) as required

This is achieved the next piece of code

```
def generate_data(n, generator_type):
ϵ_values = []
for i in range(n):
if generator_type == 'U':
e = np.random.uniform(0, 1)
else:
e = np.random.randn()
ϵ_values.append(e)
return ϵ_values
data = generate_data(100, 'U')
plt.plot(data)
plt.show()
```

Hopefully the syntax of the if/else clause is self-explanatory, with indentation again delimiting the extent of the code blocks

Notes

We are passing the argument

`U`

as a string, which is why we write it as`'U'`

Notice that equality is tested with the

`==`

syntax, not`=`

- For example, the statement
`a = 10`

assigns the name`a`

to the value`10`

- The expression
`a == 10`

evaluates to either`True`

or`False`

, depending on the value of`a`

- For example, the statement

Now, there are several ways that we can simplify the code above

For example, we can get rid of the conditionals all together by just passing the desired generator type *as a function*

To understand this, consider the following version

```
def generate_data(n, generator_type):
ϵ_values = []
for i in range(n):
e = generator_type()
ϵ_values.append(e)
return ϵ_values
data = generate_data(100, np.random.uniform)
plt.plot(data)
plt.show()
```

Now, when we call the function `generate_data()`

, we pass `np.random.uniform`

as the second argument

This object is a *function*

When the function call `generate_data(100, np.random.uniform)`

is executed, Python runs the function code block with `n`

equal to 100 and the name `generator_type`

“bound” to the function `np.random.uniform`

- While these lines are executed, the names
`generator_type`

and`np.random.uniform`

are “synonyms”, and can be used in identical ways

This principle works more generally—for example, consider the following piece of code

```
max(7, 2, 4) # max() is a built-in Python function
```

```
7
```

```
m = max
m(7, 2, 4)
```

```
7
```

Here we created another name for the built-in function `max()`

, which could then be used in identical ways

In the context of our program, the ability to bind new names to functions means that there is no problem *passing a function as an argument to another function*—as we did above

### List Comprehensions¶

We can also simplify the code for generating the list of random draws considerably by using something called a *list comprehension*

List comprehensions are an elegant Python tool for creating lists

Consider the following example, where the list comprehension is on the right-hand side of the second line

```
animals = ['dog', 'cat', 'bird']
plurals = [animal + 's' for animal in animals]
plurals
```

```
['dogs', 'cats', 'birds']
```

Here’s another example

```
range(8)
```

```
[0, 1, 2, 3, 4, 5, 6, 7]
```

```
doubles = [2 * x for x in range(8)]
doubles
```

```
[0, 2, 4, 6, 8, 10, 12, 14]
```

With the list comprehension syntax, we can simplify the lines

```
ϵ_values = []
for i in range(n):
e = generator_type()
ϵ_values.append(e)
```

into

```
ϵ_values = [generator_type() for i in range(n)]
```

## Exercises¶

### Exercise 1¶

Recall that \(n!\) is read as “\(n\) factorial” and defined as \(n! = n \times (n - 1) \times \cdots \times 2 \times 1\)

There are functions to compute this in various modules, but let’s write our own version as an exercise

In particular, write a function `factorial`

such that `factorial(n)`

returns \(n!\)
for any positive integer \(n\)

### Exercise 2¶

The binomial random variable \(Y \sim Bin(n, p)\) represents the number of successes in \(n\) binary trials, where each trial succeeds with probability \(p\)

Without any import besides `from numpy.random import uniform`

, write a function
`binomial_rv`

such that `binomial_rv(n, p)`

generates one draw of \(Y\)

Hint: If \(U\) is uniform on \((0, 1)\) and \(p \in (0,1)\), then the expression `U < p`

evaluates to `True`

with probability \(p\)

### Exercise 3¶

Compute an approximation to \(\pi\) using Monte Carlo. Use no imports besides

```
import numpy as np
```

Your hints are as follows:

- If \(U\) is a bivariate uniform random variable on the unit square \((0, 1)^2\), then the probability that \(U\) lies in a subset \(B\) of \((0,1)^2\) is equal to the area of \(B\)
- If \(U_1,\ldots,U_n\) are iid copies of \(U\), then, as \(n\) gets large, the fraction that fall in \(B\) converges to the probability of landing in \(B\)
- For a circle, area = pi * radius^2

### Exercise 4¶

Write a program that prints one realization of the following random device:

- Flip an unbiased coin 10 times
- If 3 consecutive heads occur one or more times within this sequence, pay one dollar
- If not, pay nothing

Use no import besides `from numpy.random import uniform`

### Exercise 5¶

Your next task is to simulate and plot the correlated time series

The sequence of shocks \(\{\epsilon_t\}\) is assumed to be iid and standard normal

In your solution, restrict your import statements to

```
import numpy as np
import matplotlib.pyplot as plt
```

Set \(T=200\) and \(\alpha = 0.9\)

### Exercise 6¶

To do the next exercise, you will need to know how to produce a plot legend

The following example should be sufficient to convey the idea

```
import numpy as np
import matplotlib.pyplot as plt
x = [np.random.randn() for i in range(100)]
plt.plot(x, label="white noise")
plt.legend()
plt.show()
```

Running it produces a figure like so

Now, starting with your solution to exercise 5, plot three simulated time series, one for each of the cases \(\alpha=0\), \(\alpha=0.8\) and \(\alpha=0.98\)

In particular, you should produce (modulo randomness) a figure that looks as follows

(The figure nicely illustrates how time series with the same one-step-ahead conditional volatilities, as these three processes have, can have very different unconditional volatilities.)

Use a `for`

loop to step through the \(\alpha\) values

Important hints:

If you call the

`plot()`

function multiple times before calling`show()`

, all of the lines you produce will end up on the same figure- And if you omit the argument
`'b-'`

to the plot function, Matplotlib will automatically select different colors for each line

- And if you omit the argument
The expression

`'foo' + str(42)`

evaluates to`'foo42'`

## Solutions¶

### Exercise 2¶

```
from numpy.random import uniform
def binomial_rv(n, p):
count = 0
for i in range(n):
U = uniform()
if U < p:
count = count + 1 # Or count += 1
return count
binomial_rv(10, 0.5)
```

```
6
```

### Exercise 3¶

Consider the circle of diameter 1 embedded in the unit square

Let \(A\) be its area and let \(r=1/2\) be its radius

If we know \(\pi\) then we can compute \(A\) via \(A = \pi r^2\)

But here the point is to compute \(\pi\), which we can do by \(\pi = A / r^2\)

Summary: If we can estimate the area of the unit circle, then dividing by \(r^2 = (1/2)^2 = 1/4\) gives an estimate of \(\pi\)

We estimate the area by sampling bivariate uniforms and looking at the fraction that fall into the unit circle

```
n = 100000
count = 0
for i in range(n):
u, v = np.random.uniform(), np.random.uniform()
d = np.sqrt((u - 0.5)**2 + (v - 0.5)**2)
if d < 0.5:
count += 1
area_estimate = count / n
print(area_estimate * 4) # dividing by radius**2
```

```
3.1496
```

### Exercise 4¶

```
from numpy.random import uniform
payoff = 0
count = 0
for i in range(10):
U = uniform()
count = count + 1 if U < 0.5 else 0
if count == 3:
payoff = 1
print(payoff)
```

```
1
```

### Exercise 5¶

The next line embeds all subsequent figures in the browser itself

```
α = 0.9
ts_length = 200
current_x = 0
x_values = []
for i in range(ts_length + 1):
x_values.append(current_x)
current_x = α * current_x + np.random.randn()
plt.plot(x_values)
plt.show()
```

### Exercise 6¶

```
αs = [0.0, 0.8, 0.98]
ts_length = 200
for α in αs:
x_values = []
current_x = 0
for i in range(ts_length):
x_values.append(current_x)
current_x = α * current_x + np.random.randn()
plt.plot(x_values, label=f'α = {α}')
plt.legend()
plt.show()
```