44. Simple Linear Regression Model#

and solving for \(\beta\) yields

We can now use (44.1) and (44.2) to calculate the optimal values for \(\alpha\) and \(\beta\)

df = df[['X','Y']].copy() # Original Data # Calculate the sample means x_bar = df['X'].mean() y_bar = df['Y'].mean() 

Now computing across the 10 observations and then summing the numerator and denominator

# Compute the Sums df['num'] = df['X'] * df['Y'] - y_bar * df['X'] df['den'] = pow(df['X'],2) - x_bar * df['X'] β = df['num'].sum() / df['den'].sum() print(β) 

64.37665782493369

α = y_bar - β * x_bar print(α) 

-14.72148541114052

Now we can plot the OLS solution

df['Y_hat'] = α + β * df['X'] df['error'] = df['Y_hat'] - df['Y'] fig, ax = plt.subplots() ax = df.plot(x='X',y='Y', kind='scatter', ax=ax) ax = df.plot(x='X',y='Y_hat', kind='line', ax=ax, color='g') plt.vlines(df['X'], df['Y_hat'], df['Y'], color='r'); 

Now that you know the equations that solve the simple linear regression model using OLS you can now run your own regressions to build a model between \(y\) and \(x\) .

Let’s consider two economic variables GDP per capita and Life Expectancy.

1. What do you think their relationship would be?
2. Gather some data from our world in data
3. Use pandas to import the csv formatted data and plot a few different countries of interest
4. Use (44.1) and (44.2) to compute optimal values for \(\alpha\) and \(\beta\)
5. Plot the line of best fit found using OLS
6. Interpret the coefficients and write a summary sentence of the relationship between GDP per capita and Life Expectancy

Q2: Gather some data from our world in data

You can download a copy of the data here if you get stuck

Q3: Use pandas to import the csv formatted data and plot a few different countries of interest

data_url = "https://github.com/QuantEcon/lecture-python-intro/raw/main/lectures/_static/lecture_specific/simple_linear_regression/life-expectancy-vs-gdp-per-capita.csv" df = pd.read_csv(data_url, nrows=10) 

You can see that the data downloaded from Our World in Data has provided a global set of countries with the GDP per capita and Life Expectancy Data.

It is often a good idea to at first import a few lines of data from a csv to understand its structure so that you can then choose the columns that you want to read into your DataFrame.

You can observe that there are a bunch of columns we won’t need to import such as Continent

So let’s built a list of the columns we want to import

cols = ['Code', 'Year', 'Life expectancy at birth (historical)', 'GDP per capita'] df = pd.read_csv(data_url, usecols=cols) df 

62156 rows × 4 columns

Sometimes it can be useful to rename your columns to make it easier to work with in the DataFrame

df.columns = ["cntry", "year", "life_expectancy", "gdppc"] df 

62156 rows × 4 columns

We can see there are NaN values which represents missing data so let us go ahead and drop those

df.dropna(inplace=True) 

12445 rows × 4 columns

We have now dropped the number of rows in our DataFrame from 62156 to 12445 removing a lot of empty data relationships.

Now we have a dataset containing life expectancy and GDP per capita for a range of years.

It is always a good idea to spend a bit of time understanding what data you actually have.

For example, you may want to explore this data to see if there is consistent reporting for all countries across years

Let’s first look at the Life Expectancy Data

le_years = df[['cntry', 'year', 'life_expectancy']].set_index(['cntry', 'year']).unstack()['life_expectancy'] le_years 

166 rows × 310 columns

As you can see there are a lot of countries where data is not available for the Year 1543!

Which country does report this data?

le_years[~le_years[1543].isna()] 

1 rows × 310 columns

You can see that Great Britain (GBR) is the only one available

You can also take a closer look at the time series to find that it is also non-continuous, even for GBR.

le_years.loc['GBR'].plot() 

In fact we can use pandas to quickly check how many countries are captured in each year

le_years.stack().unstack(level=0).count(axis=1).plot(xlabel="Year", ylabel="Number of countries"); 

So it is clear that if you are doing cross-sectional comparisons then more recent data will include a wider set of countries

Now let us consider the most recent year in the dataset 2018

df = df[df.year == 2018].reset_index(drop=True).copy() 

df.plot(x='gdppc', y='life_expectancy', kind='scatter', xlabel="GDP per capita", ylabel="Life expectancy (years)",); 

This data shows a couple of interesting relationships.

1. there are a number of countries with similar GDP per capita levels but a wide range in Life Expectancy
2. there appears to be a positive relationship between GDP per capita and life expectancy. Countries with higher GDP per capita tend to have higher life expectancy outcomes

Even though OLS is solving linear equations – one option we have is to transform the variables, such as through a log transform, and then use OLS to estimate the transformed variables.

By specifying logx you can plot the GDP per Capita data on a log scale

df.plot(x='gdppc', y='life_expectancy', kind='scatter', xlabel="GDP per capita", ylabel="Life expectancy (years)", logx=True); 

As you can see from this transformation – a linear model fits the shape of the data more closely.

df['log_gdppc'] = df['gdppc'].apply(np.log10)