part 17: regression residuals 17-1/38 statistics and data analysis professor william greene stern...

Part 17: Regression Residuals17-1/38

Statistics and Data Analysis

Professor William Greene

Stern School of Business

IOMS Department

Department of Economics

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Statistics and Data Analysis

Part 17 – The Linear Regression Model

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Regression Modeling

Theory behind the regression model Computing the regression statistics Interpreting the results Application: Statistical Cost Analysis

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A Linear Regression

Predictor: Box Office = -14.36 + 72.72 Buzz

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Data and Relationship

We suggested the relationship between box office sales and internet buzz is Box Office = -14.36 + 72.72 Buzz

Box Office is not exactly equal to -14.36+72.72xBuzz How do we reconcile the equation with the data?

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Modeling the Underlying Process

A model that explains the process that produces the data that we observe: Observed outcome = the sum of two parts (1) Explained: The regression line (2) Unexplained (noise): The remainder.

Internet Buzz is not the only thing that explains Box Office, but it is the only variable in the equation.

Regression model The “model” is the statement that part (1) is the

same process from one observation to the next.

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The Population Regression

THE model: (1) Explained:

Explained Box Office = α + β Buzz (2) Unexplained: The rest is “noise, ε.”

Random ε has certain characteristics Model statement

Box Office = α + β Buzz + ε Box Office is related to Buzz, but is not exactly

equal to α + β Buzz

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The Data Include the Noise

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What explains the noise?What explains the variation in fuel bills?

ROOMS

FUEL

BIL

L

111098765432

1400

1200

1000

800

600

400

200

Scatterplot of FUELBILL vs ROOMS

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Noisy Data?What explains the variation in milk production other

than number of cows?

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Assumptions

(Regression) The equation linking “Box Office” and “Buzz” is stable

E[Box Office | Buzz] = α + β Buzz

Another sample of movies, say 2012, would obey the same fundamental relationship.

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Model Assumptions

yi = α + β xi + εi α + β xi is the “regression function” εi is the “disturbance. It is the unobserved

random component The Disturbance is Random Noise

Mean zero. The regression is the mean of yi.

εi is the deviation from the regression. Variance σ2.

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We will use the data to estimate and β

Sample : a + b Buzz

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We also want to estimate 2 =√E[εi2]

Sample : a + b Buzz

e=y-a-bBuzz

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Standard Deviation of the Residuals

Standard deviation of εi = yi-α-βxi is σ

σ = √E[εi2] (Mean of εi is zero)

Sample a and b estimate α and β Residual ei = yi – a – bxi estimates εi

Use √(1/N-2)Σei2 to estimate σ.

N N2 2i i ii=1 i=1

e

e (y - a -bx )s = =

N- 2 N- 2

Why N-2? Relates to the fact that two parameters (α,β) were estimated. Same reason N-1 was used to compute a sample variance.

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Residuals

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Summary: Regression Computations

N

ii 1

N

ii 1

N2 2x ii 1

N2 2y ii 1

The same 5 statistics (with N) are still needed:

N = 62 complete observations.

1y = y = 20.721

N1

x = x = 0.48242N

1Var(x) = s = (x x) = 0.02453

N-11

Var(y) = s = (y y) = 305N-1

xy

N

i ii 1

.985

Cov(x,y) = s

1 = (x x)(y y) = 1.784

N-1

xy

2x

2 2 2y x

e

2 22 x

2y

sb = = 72.72

s

a = y - bx = -14.36

(N-1)(s -b s )s = = 13.386

N- 2(for later...),

b sR = = 0.424

s

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Using se to identify outliersRemember the empirical rule, 95% of observations will lie within mean ± 2 standard deviations? We show (a+bx) ± 2se below.)

This point is 2.2 standard deviations from the regression.

Only 3.2% of the 62 observations lie outside the bounds. (We will refine this later.)

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Linear Regression

Sample Regression Line

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Results to Report

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The Reported Results

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Estimated equation

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Estimated coefficients a and b

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S = se = estimated std. deviation of ε

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Square of the sample correlation between x and y

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N-2 = degrees of freedom

N-1 = sample size minus 1

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Sum of squared residuals, Σiei

2

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S2 = se2

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N 2ii=1

Total Variation

= (y - y)

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2

N2

N

2ii=1

2ii=1

Coefficient of Determination R

b (x - x)= =

(y - y)

RegressionSS

TotalSS

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The Model

Constructed to provide a framework for interpreting the observed data What is the meaning of the observed relationship

(assuming there is one) How it’s used

Prediction: What reason is there to assume that we can use sample observations to predict outcomes?

Testing relationships

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A Cost Model

Electricity.mpj

Total cost in $Million

Output in Million KWH

N = 123 American electric utilities

Model: Cost = α + βKWH + ε

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Cost Relationship

Output

Cost

80000700006000050000400003000020000100000

500

400

300

200

100

0

Scatterplot of Cost vs Output

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Sample Regression

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Interpreting the Model

Cost = 2.44 + 0.00529 Output + e Cost is $Million, Output is Million KWH. Fixed Cost = Cost when output = 0

Fixed Cost = $2.44Million Marginal cost

= Change in cost/change in output= .00529 * $Million/Million KWH= .00529 $/KWH = 0.529 cents/KWH.

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Summary

Linear regression model Assumptions of the model Residuals and disturbances

Estimating the parameters of the model Regression parameters Disturbance standard deviation

Computation of the estimated model