import numpy as np
import matplotlib.pyplot as plt
import pandas as pd
import scipy as sp
import statsmodels.api as sm
import pandas_datareader as pdr
import datetime as dt
from Ipython import markdown as md
Functional Forms¶
Linear regression requires parameters to be linear rather than variables, for instance an exponential form is nonlinear $$ Y_{i}=\beta_{1} X_{i}^{\beta_{2}} e^{u_{i}} $$
Many of you might wonder what if variables are nonlinear, such as $$ Y=\beta_{1}+\beta_{2} X_{2}^{3}+\beta_{3} \sqrt{X_{3}}+\beta_{4} \log X_{4} + u_i $$ Actually, it doesn't matter, they are essentially linear after nonlinear operation performed. $$ Y=\beta_{1}+\beta_{2} Z_2+\beta_{3}Z_3+\beta_{4}Z_4+u_i $$ where $Z_2 = X_{2}^{3}$, $Z_3 = \sqrt{X_{3}}$ and $Z_4 =\log X_{4}$.
But interpretation of parameters not the same anymore.
Log Form Transformation¶
We will explain how log form transformation is used through an economics example.
If you have studied microeconomics, you probably remember the concept of elasticity, it is a unitless measurement of relative changes of two variables. For example, the price elasticity of demand is defined as $$ \frac{\Delta C/C}{\Delta P/P} = \frac{d C/d P}{C/P} $$ If price elasticity of demand is $.3$, it means one percent of price change would cause $.3%$ percent change of demand.
If $C$ and $P$ takes the form of $C = \beta_1 P^{\beta_2}$, then $$ \frac{d C}{d P}=\beta_1 \beta_2 P^{\beta_2-1}= \beta_2\frac{C}{P} $$ Substitute the result back in elasticity of consumption to price $$ \frac{\Delta C/C}{\Delta P/P} = \beta_2 $$
$\beta_2$ is the elasticity, and this is also why we prefer using exponential form in estimating elasticities. Taking natural log on $C = \beta_1 P^{\beta_2}u$ $$ \ln{C} = \ln{\beta_1}+\beta_2\ln{P}+\ln{u} $$ This, again, back to linear regression form. With cosmetic substitution, the model becomes $$ C' = \beta_1'+\beta_2 P'+u' $$
Here is the estimation procedure:
- Take natural log of $C$ and $P$.
- Estimate the regression by OLS.
- $\beta_2$ has the direct interpretation of elasticity.
- To obtain $\beta_1$, take $\exp{\beta_1'}$.
We will retrieve Consumer Price Index and Expenditure of Durable Good from FRED.
start = dt.datetime(1950, 1, 1)
end = dt.datetime(2021, 1, 1)
df_exp = pdr.data.DataReader(["PCEDG", "PCE"], "fred", start, end)
df_exp.columns = ["Exp_Dur", "Tot_Exp"]
df_exp = df_exp.dropna()
(df_exp / df_exp.iloc[0]).plot(
grid=True,
figsize=(12, 7),
title="Total Exp. and Exp. on Durables (Normalised at 1.2010)",
)
plt.show()
X = np.log(df_exp["Tot_Exp"])
Y = np.log(df_exp["Exp_Dur"])
X = sm.add_constant(X) # adding a constant
model = sm.OLS(Y, X).fit()
print_model = model.summary()
print(print_model)
OLS Regression Results
==============================================================================
Dep. Variable: Exp_Dur R-squared: 0.995
Model: OLS Adj. R-squared: 0.995
Method: Least Squares F-statistic: 1.398e+05
Date: Sat, 24 Jul 2021 Prob (F-statistic): 0.00
Time: 18:20:51 Log-Likelihood: 805.70
No. Observations: 745 AIC: -1607.
Df Residuals: 743 BIC: -1598.
Df Model: 1
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
const -1.4413 0.020 -72.514 0.000 -1.480 -1.402
Tot_Exp 0.9242 0.002 373.902 0.000 0.919 0.929
==============================================================================
Omnibus: 163.607 Durbin-Watson: 0.076
Prob(Omnibus): 0.000 Jarque-Bera (JB): 35.463
Skew: -0.182 Prob(JB): 1.99e-08
Kurtosis: 1.995 Cond. No. 53.9
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
md(
"That means every $1\%$ increase of total expenditure will bring up {:.4f}\% of durable good consumption.".format(
model.params[1]
)
)
That means every $1\%$ increase of total expenditure will bring up 0.9242% of durable good consumption.