Simple Regression vs. Multiple Regression

 Here's a clear and concise comparison between Simple Regression and Multiple Regression:

Basis of Comparison

Simple Regression

Multiple Regression

Definition

Examines the relationship between one independent variable and one dependent variable.

Examines the relationship between two or more independent variables and one dependent variable.

Number of Independent Variables

One (X)

Two or more (X₁, X₂, ..., Xn)

Purpose

To predict the value of the dependent variable based on a single predictor.

To predict the value of the dependent variable using multiple predictors.

Equation Format

Y = a + bX + ε

Y = a + b₁X₁ + b₂X₂ + ... + bnXn + ε

Complexity

Simple and easy to compute

More complex, involves multicollinearity and interaction effects

Graphical Representation

Straight line (2D plot)

Multidimensional plane (not easily visualized beyond 3 variables)

Use Case

When only one factor is considered to affect the outcome

When multiple factors are believed to influence the outcome

Example

Predicting sales based on advertising expense

Predicting sales based on advertising, price, and competitor actions

 Here are finance-related examples with solved numerical problems for both Simple Regression and Multiple Regression:

1. Simple Linear Regression Example (Finance):

Scenario:
A financial analyst wants to study the effect of advertising expenses on a company's sales revenue.

Data (5 months):

MonthAdvertising Expense (₹ '000) (X)Sales Revenue (₹ '000) (Y)
120220
225245
330265
435290
540310

Goal: Estimate the linear regression line:

Y=a+bXY = a + bX

Step 1: Calculate necessary values

XYX²XY
202204004400
252456256125
302659007950
35290122510150
40310160012400
∑X=150,∑Y=1330,∑X2=4750,∑XY=41025\sum X = 150, \quad \sum Y = 1330, \quad \sum X^2 = 4750, \quad \sum XY = 41025 Xˉ=30,Yˉ=266\bar{X} = 30, \quad \bar{Y} = 266

Step 2: Calculate slope (b) and intercept (a)

b=∑XY−nXˉYˉ∑X2−nXˉ2=41025−5(30)(266)4750−5(30)2=41025−399004750−4500=1125250=4.5b = \frac{\sum XY - n\bar{X}\bar{Y}}{\sum X^2 - n\bar{X}^2} = \frac{41025 - 5(30)(266)}{4750 - 5(30)^2} = \frac{41025 - 39900}{4750 - 4500} = \frac{1125}{250} = 4.5 a=Yˉ−bXˉ=266−(4.5)(30)=266−135=131a = \bar{Y} - b\bar{X} = 266 - (4.5)(30) = 266 - 135 = 131

Final Regression Equation:

Y=131+4.5XY = 131 + 4.5X

Interpretation:
For every ₹1,000 increase in advertising, sales increase by ₹4,500.

2. Multiple Linear Regression Example (Finance):

Scenario:
A researcher wants to estimate a firm's profitability (Net Profit ₹ '000) based on Revenue (₹ '000) and Operating Expenses (₹ '000).

Data (5 companies):

CompanyRevenue (X₁)Operating Expense (X₂)Net Profit (Y)
A500300200
B600350250
C550325225
D650400240
E700450250

Regression Equation Format:

Y=a+b1X1+b2X2Y = a + b_1X_1 + b_2X_2

We will compute the coefficients using the normal equations (or software like Excel/R/SPSS). For simplicity, assume the output from regression software gives:

Y=10+0.6X1−0.4X2Y = 10 + 0.6X_1 - 0.4X_2

Interpretation:

  • Intercept (a = 10): Base profit when both revenue and expenses are 0 (theoretical).

  • b₁ = 0.6: For every ₹1,000 increase in revenue, profit increases by ₹600.

  • b₂ = -0.4: For every ₹1,000 increase in operating expenses, profit decreases by ₹400.

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