Exercises: Instability of Parameter Estimates

This exercise notebook refers to this lecture. Please use the lecture for explanations and sample code.

https://www.quantopian.com/lectures#Instability-of-Estimates

Part of the Quantopian Lecture Series:

In [ ]:
import numpy as np
import matplotlib.pyplot as plt
import pandas as pd
from statsmodels.stats.stattools import jarque_bera

# Set a seed so we can play with the data without generating new random numbers every time

Exercise 1: Sample Size vs. Standard Deviation

Using the below normal distribution with mean 100 and standard deviation 50, find the means and standard deviations of samples of size 5, 25, 100, and 500.

In [ ]:
POPULATION_MU = 100
POPULATION_SIGMA = 25
sample_sizes = [5, 25, 100, 500]

#Your code goes here

Exercise 2: Instability of Predictions on Mean Alone

a. Finding Means

Find the means of the following three data sets $X$, $Y$, and $Z$.

In [ ]:
X = [ 31.,   6.,  21.,  32.,  41.,   4.,  48.,  38.,  43.,  36.,  50., 20.,  46.,  33.,   8.,  27.,  17.,  44.,  16.,  39.,   3.,  37.,
        35.,  13.,  49.,   2.,  18.,  42.,  22.,  25.,  15.,  24.,  11., 19.,   5.,  40.,  12.,  10.,   1.,  45.,  26.,  29.,   7.,  30.,
        14.,  23.,  28.,   0.,  34.,   9.,  47.]
Y = [ 15.,  41.,  33.,  29.,   3.,  28.,  28.,   8.,  15.,  22.,  39., 38.,  22.,  10.,  39.,  40.,  24.,  15.,  21.,  25.,  17.,  33.,
        40.,  32.,  42.,   5.,  39.,   8.,  15.,  25.,  37.,  33.,  14., 25.,   1.,  31.,  45.,   5.,   6.,  19.,  13.,  39.,  18.,  49.,
        13.,  38.,   8.,  25.,  32.,  40.,  17.]
Z = [ 38.,  23.,  16.,  35.,  48.,  18.,  48.,  38.,  24.,  27.,  24., 35.,  37.,  28.,  11.,  12.,  31.,  -1.,   9.,  19.,  20.,   0.,
        23.,  33.,  34.,  24.,  14.,  28.,  12.,  25.,  53.,  19.,  42., 21.,  15.,  36.,  47.,  20.,  26.,  41.,  33.,  50.,  26.,  22.,
        -1.,  35.,  10.,  25.,  23.,  24.,   6.]

#Your code goes here

b. Checking for Normality

Use the jarque_bera function to conduct a Jarque-Bera test on $X$, $Y$, and $Z$ to determine whether their distributions are normal.

In [ ]:
#Your code goes here

c. Instability of Estimates

Create a histogram of the sample distributions of $X$, $Y$, and $Z$ along with the best estimate/mean based on the sample.

In [ ]:
#Your code goes here

Exercise 3: Sharpe Ratio Window Adjustment

a. Effect on Variability

Just as in the lecture, find the mean and standard deviation of the running sharpe ratio for THO, this time testing for multiple window lengths: 300, 150, and 50. Restrict your mean and standard deviation calculation to pricing data up to 200 days away from the end.

In [ ]:
def sharpe_ratio(asset, riskfree):
    return np.mean(asset - riskfree)/np.std(asset - riskfree)

start = '2010-01-01'
end = '2015-01-01'

treasury_ret = get_pricing('BIL', fields='price', start_date=start, end_date=end).pct_change()[1:]
pricing = get_pricing('THO', fields='price', start_date=start, end_date=end)
returns = pricing.pct_change()[1:]

#Your code goes here

b. Out-of-Sample Instability

Plot the running sharpe ratio of all three window lengths, as well as their in-sample mean and standard deviation bars.

In [ ]:
#Your code goes here

Exercise 4: Weather

a. Temperature in Boston

Find the mean and standard deviation of Boston weekly average temperature data for the year of 2015 stored in b15_df.

In [ ]:
b15_df = pd.DataFrame([ 29.,  22.,  19.,  17.,  19.,  19.,  15.,  16.,  18.,  25.,  21.,
        25.,  29.,  27.,  36.,  38.,  40.,  44.,  49.,  50.,  58.,  61.,
        67.,  69.,  74.,  72.,  76.,  81.,  81.,  80.,  83.,  82.,  80.,
        79.,  79.,  80.,  74.,  72.,  68.,  68.,  65.,  61.,  57.,  50.,
        46.,  42.,  41.,  35.,  30.,  27.,  28.,  28.],
        columns = ['Weekly Avg Temp'],
        index = pd.date_range('1/1/2012', periods=52, freq='W')          )

#Your code goes here

b. Temperature in Palo Alto

Find the mean and standard deviation of Palo Alto weekly average temperature data for the year of 2015 stored in p15_df.

In [ ]:
p15_df = pd.DataFrame([ 49.,  53.,  51.,  47.,  50.,  46.,  49.,  51.,  49.,  45.,  52.,
        54.,  54.,  55.,  55.,  57.,  56.,  56.,  57.,  63.,  63.,  65.,
        65.,  69.,  67.,  70.,  67.,  67.,  68.,  68.,  70.,  72.,  72.,
        70.,  72.,  70.,  66.,  66.,  68.,  68.,  65.,  66.,  62.,  61.,
        63.,  57.,  55.,  55.,  55.,  55.,  55.,  48.],
        columns = ['Weekly Avg Temp'],
        index = pd.date_range('1/1/2012', periods=52, freq='W'))

#Your code goes here

c. Predicting 2016 Temperatures

Use the means you found in parts a and b to attempt to predict 2016 temperature data for both cities. Do this by creating two histograms for the 2016 temperature data in b16_df and p16_df with a vertical line where the 2015 means were to represent your prediction.

In [ ]:
b16_df = pd.DataFrame([ 26.,  22.,  20.,  19.,  18.,  19.,  17.,  17.,  19.,  20.,  23., 22.,  28.,  28.,  35.,  38.,  42.,  47.,  49.,  56.,  59.,  61.,
        61.,  70.,  73.,  73.,  73.,  77.,  78.,  82.,  80.,  80.,  81., 78.,  82.,  78.,  76.,  71.,  69.,  66.,  60.,  63.,  56.,  50.,
        44.,  43.,  34.,  33.,  31.,  28.,  27.,  20.],
        columns = ['Weekly Avg Temp'],
        index = pd.date_range('1/1/2012', periods=52, freq='W'))

p16_df = pd.DataFrame([ 50.,  50.,  51.,  48.,  48.,  49.,  50.,  45.,  52.,  50.,  51., 52.,  50.,  56.,  58.,  55.,  61.,  56.,  61.,  62.,  62.,  64.,
        64.,  69.,  71.,  66.,  69.,  70.,  68.,  71.,  70.,  69.,  72., 71.,  66.,  69.,  70.,  70.,  66.,  67.,  64.,  64.,  65.,  61.,
        61.,  59.,  56.,  53.,  55.,  52.,  52.,  51.],
        columns = ['Weekly Avg Temp'],
        index = pd.date_range('1/1/2012', periods=52, freq='W'))

#Your code goes here

Congratulations on completing the instability of parameter estimates exercises!

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