Introduction

Linear Regression is a supervised learning algorithm used to model the relationship between a dependent variable (target) and one or more independent variables (predictors). It assumes a linear relationship between the variables and aims to find the best-fit line that minimizes the difference between actual and predicted values. There two types of Linear Regression:

• Simple Linear Regression (One predictor)
• Multiple Linear Regression (Multiple predictors)

Importance in Financial Analytics

Assumptions of Simple Linear Regression

1. Linearity: The relationship between X and y is linear.
2. Independence: Observations are independent of each other.
3. Normality: The residuals are normally distributed.(Normality of error distribution)
4. Homoscedasticity: The residuals have constant variance across all levels of X(Equal variance)
5. No Multicollinearity: Only one predictor is used, so multicollinearity is not applicable.

Assumptions of Linear Regression (Remember: LINE)

Data used for linear regression must follow a linear pattern. Among the patterns shown below, we can only apply linear regression to the first one, not the others..

Feature Selection & Engineering

• Handling categorical variables (One-Hot Encoding, Label Encoding)
• Creating interaction terms between features
• Scaling numerical variables (Standardization & Normalization)
• Detecting and treating multicollinearity (Variance Inflation Factor - VIF)
• Feature selection techniques:
• Backward Elimination
• Forward Selection
• Stepwise Regression
• Lasso & Ridge Regression (Regularization)

Simple Linear Regression (One Predictor)

Multiple Linear Regression (Multiple Predictors)

Difference Between Prediction and Inference in Linear Regression (ML)

In machine learning, particularly in linear regression, we distinguish between prediction and inference:

• Prediction: Using the trained model to make future predictions based on new, unseen data.
• Inference: Understanding the relationships between the independent variables (features) and the dependent variable (target).

Case Study: House Price Prediction

Let’s consider a real estate company that wants to analyze house prices.

• Prediction Use Case: Given the size of a house, predict its price.
• Inference Use Case: Understanding how much the size of a house contributes to price increases.

Step 1: Load and Explore Data

We'll simulate a dataset where the house price depends on size (sqft) and number of bedrooms.

python
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import numpy as np
import pandas as pd
import statsmodels.api as sm
from sklearn.model_selection import train_test_split
from sklearn.linear_model import LinearRegression

# Generate synthetic dataset
np.random.seed(42)
size = np.random.randint(1000, 3000, 100)  # House size in sqft
bedrooms = np.random.randint(1, 5, 100)   # Number of bedrooms
price = 50000 + (size * 150) + (bedrooms * 10000) + np.random.normal(0, 10000, 100)  # House price

# Create DataFrame
df = pd.DataFrame({'Size': size, 'Bedrooms': bedrooms, 'Price': price})

# Split into train and test sets
X = df[['Size', 'Bedrooms']]
y = df['Price']
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)

Step 2: Train a Linear Regression Model

We will fit a linear regression model to predict house prices.

# Train model
model = LinearRegression()
model.fit(X_train, y_train)

# Get model coefficients
print(f"Intercept: {model.intercept_}")
print(f"Coefficients: {dict(zip(X.columns, model.coef_))}")

Prediction Example

Now, let’s use the model to predict house prices for new houses.

python
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# New house details
new_house = pd.DataFrame({'Size': [2500, 1800], 'Bedrooms': [3, 2]})

# Predict house prices
predicted_prices = model.predict(new_house)
print(f"Predicted Prices: {predicted_prices}")

Key Insight:

• We are using the trained model to estimate the price of new, unseen houses.
• This is purely predictive since we do not analyze the relationship between features and the target.

Inference Example

To understand the impact of each feature on the house price, we use Statistical Inference.

python
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# Add constant term for intercept in statsmodels
X_train_sm = sm.add_constant(X_train)

# Fit the OLS regression model
ols_model = sm.OLS(y_train, X_train_sm).fit()

# Display summary statistics
print(ols_model.summary())

Key Insight:

• The p-values help determine if a variable significantly impacts price.
• The coefficients tell us how much price changes per unit increase in each feature.
• This is useful for making business decisions, such as whether adding a bedroom has a significant financial impact.

Final Takeaway

Both prediction and inference are crucial in machine learning, but their use cases differ. Prediction is useful for forecasting, while inference helps explain and understand model relationships.

                          OLS Regression Results                            
==============================================================================
Dep. Variable:                  Price   R-squared:                       0.990
Model:                            OLS   Adj. R-squared:                  0.990
Method:                 Least Squares   F-statistic:                     3945.
Date:                Thu, 20 Mar 2025   Prob (F-statistic):           2.70e-78
Time:                        17:25:32   Log-Likelihood:                -838.07
No. Observations:                  80   AIC:                             1682.
Df Residuals:                      77   BIC:                             1689.
Df Model:                           2                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
const       4.094e+04   4410.541      9.281      0.000    3.22e+04    4.97e+04
Size         153.4489      1.735     88.441      0.000     149.994     156.904
Bedrooms    1.112e+04    875.934     12.699      0.000    9378.960    1.29e+04
==============================================================================
Omnibus:                        1.088   Durbin-Watson:                   2.034
Prob(Omnibus):                  0.581   Jarque-Bera (JB):                0.751
Skew:                           0.234   Prob(JB):                        0.687
Kurtosis:                       3.080   Cond. No.                     9.51e+03
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
[2] The condition number is large, 9.51e+03. This might indicate that there are
strong multicollinearity or other numerical problems.

Understanding OLS Regression Results in Financial Data Analysis

When working with Ordinary Least Squares (OLS) Regression, interpreting the results correctly is crucial for making data-driven financial decisions. Let's break down each component of the regression output and understand its significance, along with how you can apply it as a financial data analyst.

1. Model Summary

This section provides the overall statistics for the model.

2. Model Performance Metrics

These metrics assess the goodness of fit of the model.

3. Regression Coefficients

These values represent the impact of each predictor on the dependent variable.

4. Statistical Significance of Coefficients

These help us determine which variables significantly impact price.

5. Model Diagnostics

These tests validate model assumptions.

How to Apply These Insights as a Financial Data Analyst

• Investment Decision-Making: Use regression analysis to predict stock prices, revenue, sales forecasts, and investment returns.
• Risk Assessment: Identify which factors (e.g., interest rates, GDP) significantly impact financial markets.
• Pricing Strategy: Optimize real estate pricing, loan pricing, or insurance premiums using coefficient interpretations.
• Forecasting: Develop predictive models for sales, demand, or expenses based on historical data.
• Multicollinearity Handling: If the condition number is too high, use variance inflation factor (VIF) or principal component analysis (PCA) to remove redundancy.
• Model Validation: Ensure assumptions like normality, no autocorrelation, and significant predictors hold before making financial decisions.

Default Values & How to Modify Them

Final Thoughts

Understanding OLS regression output is essential for making data-driven financial decisions. As a financial data analyst, mastering these metrics allows you to:

1. Build accurate predictive models.
2. Interpret key drivers of financial trends.
3. Ensure model validity for business applications.

Would you like me to demonstrate how to apply this on real financial data (e.g., stock prices or sales forecasting)? 🚀

You said:

Do I need to calculate inference if I use regression or polynomial to check

ChatGPT said:

Do You Need Inference When Using Regression or Polynomial Regression?

It depends on your goal:

1. If your goal is prediction, you don’t necessarily need inference.
• Example: Forecasting future sales or stock prices using polynomial regression.
• You focus on making accurate predictions, not understanding relationships.
2. If your goal is understanding the relationships between variables (causation/inference), then you must perform inference.
• Example: Evaluating how much sales increase when marketing spend increases.
• You analyze coefficients, confidence intervals, and significance (p-values, F-tests).

Regression Types and When to Use Inference

Case Study: When Inference is NOT Needed

Let's say we are predicting house prices using polynomial regression.

1. Build a Polynomial Regression Model (Only for Prediction)

python
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from sklearn.preprocessing import PolynomialFeatures
from sklearn.linear_model import LinearRegression
from sklearn.pipeline import make_pipeline
import numpy as np

# Generate synthetic data
np.random.seed(42)
X = np.random.randint(1000, 3000, 100).reshape(-1, 1)  # House size
y = 50000 + (X.flatten()**2 * 0.00005) + np.random.normal(0, 5000, 100)  # Price

# Train polynomial regression model
poly_model = make_pipeline(PolynomialFeatures(degree=2), LinearRegression())
poly_model.fit(X, y)

# Predict for a new house
new_house = np.array([[2500]])
predicted_price = poly_model.predict(new_house)
print(f"Predicted Price: {predicted_price}")

Why We Skip Inference?

• Polynomial regression coefficients are not easily interpretable.
• We don't care about the impact of each variable; we only care about the prediction.

Case Study: When Inference is Needed

Now, let's check if size significantly impacts house prices.

2. Perform Inference Using OLS

python
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import statsmodels.api as sm

# Add constant term for intercept
X_sm = sm.add_constant(X)

# Fit an OLS model
ols_model = sm.OLS(y, X_sm).fit()

# Display summary
print(ols_model.summary())

Why We Use Inference Here?

• We check p-values to confirm if Size significantly affects Price.
• We analyze the coefficients to quantify the effect.

Key Takeaways

If your focus is on predictions only, you don’t need inference. But if you want to explain relationships, inference is essential. 🚀

Here’s a detailed comparison table that breaks down Prediction vs. Inference under Machine Learning (ML) and Econometrics, highlighting both overlaps and distinctions:

Summary:

• Prediction (ML): Prioritizes what happens — future outcomes, recommendations, classifications.
• Inference (Econometrics): Focuses on why it happens — significance, causality, structural understanding.

They overlap in using models like linear regression and need for clean data, but they diverge in goals, assumptions, tools, and interpretation.

Would you like this formatted as a downloadable table or integrated into a study note format?