Extra Bits and Bytes for Regression

Business Analytics

Interpreting Coefficients

I think it's important to keep in mind that we are fitting a linear model to data without assuming the the Conditional Expectation Function is linear

Y_i = \beta_0 + \beta_1X_i + \varepsilon_i

Linear Regression Model

The average value of Y when X is zero

Interpretation

The predicted value of Y when X is zero

More Precise Interpretation

Y_i = \beta_0 + \beta_1X_i + \varepsilon_i

Linear Regression Model

The average change in Y when X increases by one-unit

Interpretation

The predicted change in Y when X increase by one Unit

More Precise Interpretation

In the following linear regression model, what does

\beta_0, \beta_1

(A)

(B)

capture, respectively?

\text{Price}_i = \beta_0 + \beta_1 \text{Size}_i + \varepsilon_i

The average price of a house, The average change in price when the size of the house increases by one unit

The average price of a house when size is zero, the average change in price when the size of the house increases by one unit

(C)

The fixed cost of a house, the marginal cost of the house

The average price of a house when size is zero, the average effect of increasing the size of the house by one unit

(D)

Y_i = \beta_0 + \beta_1X_{1i} + \beta_2X_{2i}+ \varepsilon_i

Linear Regression Model

The average change in Y when X1 increases by one-unit holding X2 Constant

Interpretation

The predicted change in Y when X1 increase by one unit holding X2 constant

More Precise Interpretation

In the following linear regression model, what does

\beta_0, \beta_1

(A)

capture, respectively?

\text{Price}_i = \beta_0 + \beta_1 \text{Size}_i + \beta_2\text{Beacon}_i + \varepsilon_i

The average price of a house when Beacon is zero, The average change in price when the size of the house increases by one unit holding Beacon constant

The Fixed Cost, the Marginal Cost

(B)

(D)

The average price of a house when Size is zero, The average change in price when the size of the house increases by one unit holding Beacon constant

(C)

The average price of a house when Size and Beacon is zero, The average change in price when the size of the house increases by one unit holding Beacon constant

Y_i = \beta_0 + \beta_1X_{i} + \varepsilon_i

Linear Regression Model with a Dummy Variable

The average value of Y when X is zero

Interpretation

The difference in Average Value of Y when X is 0 and when X is 1

Interpretation

X_i \in \{0, 1\}

Dummy Variable

In the following linear regression model, what does

\beta_2

capture

\text{Price}_i = \beta_0 + \beta_1 \text{Size}_i + \beta_2\text{Beacon}_i + \varepsilon_i

In the following linear regression model, what does

\beta_3

(A)

(B)

capture?

\text{Price}_i = \beta_0 + \beta_1\text{Size}_i + \beta_2 \text{Beacon}_i + \beta_3\text{Size}_i \times \text{Beacon}_i + \beta_4 \text{garage}_i + \varepsilon_i

The effect of increasing the size of a house by one unit on the price of the house

(C)

The difference between Beacon and non-Beacon areas in the average change in Price associated with a one-unit increase in Size holding Garage constant

The average change in Price when increasing Size by one unit holding Beacon and Garage constant

In the following linear regression model, what does

\beta_0

(A)

(B)

capture?

\text{Price}_i = \beta_0 + \text{C(BuildingStyle)}_i + \varepsilon_i

The average price of a house

(C)

The average price of a house in the reference/default category of buildingStyle

The average price of a house with no building style

More Questions

linear_model = smf.ols('price ~ size + beacon', data=df)

Which linear regression equation corresponds to the following line of code?

(A) 

(B) 

(C) 

\text{Size}_i = \beta_0 + \beta_1\text{Price}_i + \text{Beacon}_i + \varepsilon_i
\text{Price}_i = \beta_1\text{Size}_i + \beta_2\text{Beacon}_i + \varepsilon_i
\text{Price}_i = \beta_0 + \beta_1\text{Size}_i + \beta_2\text{Beacon}_i

(D) 

\text{Price}_i = \beta_0 + \beta_1\text{Size}_i + \beta_2\text{Beacon}_i + \varepsilon_i
\text{Price}_i = \beta_0 + \beta_1\text{Size}_i + \beta_2\text{Size}^2_i + \varepsilon_i

What is the best interpretation of 

\beta_2

in the following Quadratic Regression?

(A) 

(B) 

(C) 

(D) 

The average change in price when increasing size by one unit

We cannot include higher order terms in a linear regression model

The average change in price when increasing the squared size by one unit, holding the size constant

Captures how the marginal change in price with respect to size varies as we increase size

More Questions

Which linear model will have the lowest Mean Squared Error?

(A) 

linear_modelA = smf.ols('price ~ size + I(size **2)', data=df)

(B) 

linear_modelB = smf.ols('price ~ size + I(size **2) + I(size**3)', data=df)

(C) 

linear_modelC = smf.ols('price ~ size', data=df)

True or False

(A) 

(B) 

True

False

A low 

is indicative of selection bias

R^2