# Lesson 18

## Logic and Loops¶

This lesson covers:

• Mixing logic and loops
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# Setup: Load the momentum data

import pandas as pd

mom_01 = momentum.mom_01

### Problem: Logical Statements and for Loops¶

Use a for loop along with an if statement to simulate an asymmetric random walk of the form

$$y_{i}=y_{i-1}+e_{i}+I_{[e_{i}<0]}e_{i}$$

where $I_{[e_{i}<0]}$ is known as an indicator variable that takes the value 1 if the statement in brackets is true. Plot y. $e$ is a standard normal shock. Use cumsum to simulate a symmetric one (z), and plot the two using the code in the cell below.

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Plot the two random walks using the code. We will cover data visualization in a later lesson.

%matplotlib inline
import matplotlib.pyplot as plt
plt.plot(y)
plt.plot(z)
plt.legend(["y", "z"])
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### Problem: Simulate the asymmetric random walk without an if-then¶

Use boolean multiplication to simulate the same random walk without using an if-then statement.

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# Setup: Plot the data
%matplotlib inline
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### Problem: Combining flow control¶

For momentum portfolios 1 and 10, compute the length of the runs in the series. In pseudo code,

• Start at i=1 and define run(1) = 1
• For i in 2,...,T, define run(i) = run(i-1) + 1 if $\textrm{sgn}\left(r_{i}\right)=\textrm{sgn}\left(r_{i-1}\right)$ else 1.

You will need to use len and zeros.

1. Compute the length longest run in the series and the index of the location of the longest run. Was it positive or negative?
2. How many distinct runs lasted 5 or more days?
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Plot the runs using

%matplotlib inline

import matplotlib.pyplot as plt
plt.plot(run)
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### Exercises¶

#### Exercise: Simulate a Process with Heteroskedasticity¶

Simulate 100 observations of a time series with heteroskedasticity that follows a random walk of the form:

$$y_t = y_{t-1} + \sigma_t \epsilon_t$$

where $\epsilon_t\sim N(0,1)$, $y_0=0$ and $\sigma_t$ is:

• 0.5 if the 0 of the past 3 shocks are negative
• 1 if 1 of the past 3 shocks are negative
• 2 if 2 of the past 3 shocks are negative
• 6 if 3 of the past 3 shocks are negative

Plot the result.

Notes

• When generating the first 3 values, treat $\epsilon_{-1}$, $\epsilon_{-2}$ and $\epsilon_{-3}$ as 0 (non-negative).
• Re-run the simulation to see different paths.
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