Multiple Linear RegressionReview

Outline Outline Outline

• Simple Linear Regression Simple Linear Regression

• Multiple Regression Multiple Regression

• Understanding the Regression Output Understanding the Regression Output

• Coefficient of Determination R Coefficient of Determination R2

• Validating the Regression Model

Linear Regression: An Example

Firs

t Y

ear

Sal

es (

$Mill

ions

)Advertising Expenditures ($Millions)

A pplegloFirst-YearAdvertisingExpenditures($ m illions)

First-YearSales

($ millions )Reg ion

MaineNew HampshireVermontMassachusettsConnecticutRhode IslandNew YorkNew JerseyPennsylvaniaDelawareMarylandWest VirginiaVirginiaOhio

Questions: a) How to relate advertising expenditure to sales? b) What is expected first-year sales if advertising expenditure is $2.2 million? c) How confident is your estimate? How good is the “fit?”

The Basic Model: Simple Linear Regression

Data: (x1, y1), (x2, y2), . . . , (xn, yn)Model of the population : Yi = β0 + β1 xi + εi

ε1, ε2, . . . , εn are are i.i.d. random variables, random variables, N(0, σ ) This is the true relation between Y and x, but we do not know not know β0 and β1 and have to estimate them based on the data.

Comments:• E (Yi | xi ) = β0 + β1xi

• SD(Yi | xi) = σ• Relationship is linear Relationship is linear – described by a “line” • β0 = “baseline” value of Y (i.e., value of Y if x is 0) • β1 = “slope” of line (average change in Y per unit change in x)

How do we choose the line that “best” fits the data?

Best choices:bo = 13.82 b1 = 48.60

Advertising Expenditures ($M)

Firs

t Y

ear

Sal

es (

$M)

Regression coefficients : b0 and b1 are are estimates of β0 and β1

Regression estimate for Y at xi : (prediction)

Residual (error):The “best” regression line is the one that chooses b0 and b1 to minimize the total errors (residual sum of squares):

Example: Sales of Nature-Bar ($ million)

region sales advertising promotions competitor’s sales

SelkirkSusquehannaKitteryActon Finger LakesBerkshireCentralProvidenceNashuaDunsterEndicottFive-TownsWaldeboroJacksonStowe

Multiple Regression

• In general, there are many factors in addition to advertising expenditures that affect sales • Multiple regression allows more than one x variables.Independent variables: x1, x2, . . . , xk (k of them)Data: (y1, x11, x21, . . . , xk1), . . . , (yn, x1n, x2n, . . . , xkn),Population Model: Yi = β0 + β1x1i + . . . + βkxki + εi

ε1, ε2, . . . , εn are are iid random variables, ~ N(0, σ)

Regression coefficients : b0, b1,…, bk are estimates of β0, β1,…,βk .Regression Estimate of yi :Goal: Choose b0, b , b1, ... , , ... , bk to minimize the residual sum of squares. I.e., minimize:

Regression Output (from Excel)Regression Statistics

Multiple R R Square Adjusted R Square Standard Error Observations

Analysis ofVariance

df Sum ofSquares

MeanSquare

F SignificanceF

Regression ResidualTotal

Coefficients StandardError

tStatistic

P-value

Lower95%

Upper95%

Intercept Advertising Promotions Competitor’sSales

Understanding Regression Output

1) Regression coefficients : b0, b1, . . . , bk are estimates are estimates of β0, β1, . . . , βk based on sample data. Fact: E[ Fact: E[bj ] = βj .

Example: b0 = 65.705 (its interpretation is context dependent .

b1 = 48.979 (an additional $1 million in advertising is expected to result in an additional $49 million in sales)

b2 = 59.654 (an additional $1 million in promotions is expected to result in an additional $60 million in sales)b3 = -1.838 (an increase of $1 million in competitor sales is expected to decrease sales by $1.8 million)

Understanding Regression Output, Continued

2) Standard errors : an estimate of σ, the SD of each the SD of each εi. It is a measure of the amount of “noise” in the model. Example: s = 17.60

3) Degrees of freedom : #cases - #parameters, relates to over relates to over-fitting phenomenon

4) Standard errors of the coefficients: sb0 , sb1, . . . , sbk

They are just the standard deviations of the estimates b0, b1, . . . , bk.

They are useful in assessing the quality of the coefficient estimates and validating the model.

R2 takes values between 0 and 1 (it is a percentage).

R2 = 0; x values account for none variation in the Y values

R2 = 1; x values account for all variation in the Y values

R2 = 0.833 in ourAppleglo Example

Understanding Regression Output, Continued

5) Coefficient of determination: R2

• It is a measure of the overall quality of the regression. • Specifically, it is the percentage of total variation exhibited in the yi data that is accounted for by the sample regression line. The sample mean of Y:

Total variation in Y Total variation in Y =

Residual (unaccounted) variation in Y

total variation

variation accounted for by x variables

total variation

variation not accounted for by x variables

Coefficient of Determination: R2

• A high R2 means that most of the variation we observe in the yi data can be attributed to their corresponding x values −− a desired property.

• In simple regression, the R2 is higher if the data points are is better aligned along a line. But outliers better aligned– Anscombe example.• How high a R2 is “good” enough depends on the situation (for example, the intended use of the regression, and complexity of the problem).

• Users of regression tend to be fixated on R2, but it’s not the , whole story. It is important that the regression model is “valid.”

Coefficient of Determination: R2

• One should not include x variables unrelated to Y in the model, just to make the R2 fictitiously high. (With more x variables fictitiously high. there will be more freedom in choosing the bi’s ’s to make the residual variation closer to 0).

• Multiple R is just the square root of R Multiple R is just the square root

Validating the Regression Model

Assumptions about the population: about the population: Yi = β0 + β1x1i + . . . + βkxki + εi (i = 1, . . . , n) ε1, ε2, . . . , εn are are iid random variables, ~ N(0, σ)

1) Linearity • If k = 1 (simple regression), one can check visually from scatter plot.

• “Sanity check:” the sign of the coefficients, reason for non-linearity?

2) Normality of Normality of εi

• Plot a histogram of the residuals• Usually, results are fairly robust with respect to this assumption.

3) Heteroscedasticity• Do error terms have constant Std. Dev.? (i.e., SD(εi) = σ for all i?) • Check scatter plot of residuals vs. Y and x variables.

Residuals

Advertising Expenditures

Residuals

Advertising

Residu

No evidence of heteroscedasticity Evidence of heteroscedasticity

• May be fixed by introducing a transformation • May be fixed by introducing or eliminating some independent variables

4) Autocorrelation : Are error terms independent? Plot residuals in order and check for patterns

Time Plot Time Plot

Res

idua

l

Res

idua

l

No evidence of autocorrelation Evidence of autocorrelation

• Autocorrelation may be present if observations have a naturalsequential order (for example, time).• May be fixed by introducing a variable or transforming a variable.

Pitfalls and Issues1) Overspecification • Including too many x variables to make R2 fictitiously high. • Rule of thumb: we should maintain that Rule of thumb: n >= 5(k+2).

2) Extrapolating beyond the range of data

Advertising

Validating the Regression Model

3) Multicollinearity • Occurs when two of the x variable are strongly correlated. • Can give very wrong estimates for βi’s.• Tell-tale signs: - Regression coefficients (bi’s) have the “wrong” sign. - Addition/deletion of an independent variable results in large changes of regression coefficients - Regression coefficients (bi’s ) not significantly different from 0 • May be fixed by deleting one or more independent variables

ExampleStudent Graduate CollegeNumber GPA GPA GMAT

Regression Output

What happened?

College GPA and GMAT are highly correlated!

Eliminate GMAT

R SquareStandard ErrorObservations

InterceptCollege GPA GMAT

Coefficients Standard Error

Graduate College GMAT

GraduateCollege GMAT

R Square Standard Error Observations

InterceptCollege GPA

Coefficients Standard Error

Regression Models

• In linear regression, we choose the “best” coefficients b0, b1, ... , bk as the estimates for β0, β1,…, βk .

• We know on average each bj hits the right target βj .

• However, we also want to know how confident we are aboutour estimates

Back to Regression Output

Intercept Advertising Promotions Compet.Sales

Upper95%

Lower95%

P-value

tStatistic

StandardError

Coefficients

Regression ResidualTotal

dfSum ofSquares

MeanSquare

Regression Statistics

Multiple R R Square Adjusted R Square Standard Error Observations Analysis of Variance

Regression Output Analysis

1) Degrees of freedom (dof) • Residual dof = n = n - (k+1) (We used up (k + 1) degrees of freedom in forming (k+1) sample estimates b0, b1, . . . , bk .)

2) Standard errors of the coefficients : sb0 , sb1, . . . , sbk

• They are just the SDs of estimates of b0, b1, . . . , bk .

• Fact: Before we observe bj and sbj, obeys a t-distribution with dof = (n - k - 1), the same dof as the residual.

• We will use this fact to assess the quality of our estimates bj . • What is a 95% confidence interval for βj? • Does the interval contain 0? Why do we care about this?

3) t-Statistic:

• A measure of the statistical significance of each individual xj

in accounting for the variability in Y.

• Let c be that number for which

where T obeys a t-distribution distribution with dof = (n - k - 1).

• If > c, then the c, then the α% C.I. for βj does not contain zero

• In this case, we are α% confident that confident that βj different from zero.

Example: Executive Compensation

Pay Years in Change in Change inNumber ($1,000) position Stock Price (%) Sales (%) MBA?

variables: Dummy

• Often, some of the explanatory variables in a regression are categorical rather than numeric.

• If we think whether an executive has an MBA or not affects his/her pay, We create a dummy variable and let it be 1 if the executive has an MBA and 0 otherwise.

• If we think season of the year is an important factor to determinesales, how do we create dummy variables? How many?

• What is the problem with creating 4 dummy variables?

• In general, if there are m categories an x variable can belong to,

then we need to create m-1 dummy variables for it.

OILPLUS data

August, 1989September, 1989October, 1989November, 1989December, 1989January, 1990February, 1990March, 1990April, 1990May, 1990June, 1990July, 1990August, 1990September, 1990October, 1990November, 1990December, 1990

Month heating oil temperature

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Average Temperature (degrees Fahrenheit)

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inverse temperature

heating oil temperature inverse temperature

The Practice of Regression

• Choose which independent variables to include in the model, based on common sense and context specific knowledge.

• Collect data (create dummy variables in necessary).

• Run regression Run regression −− −− the easy part.

• Analyze the output and make changes in the model −− this is where the action is.

• Test the regression result on “out-of of-sample” data

The Post-Regression Checklist1) Statistics checklist: Calculate the correlation between pairs of x variables −− watch for evidence of multicollinearity Check signs of coefficients – do they make sense? Check 95% C.I. (use t-statistics as quick scan) – are coefficients significantly different from zero? R2 :overall quality of the regression, but not the only measure

2) Residual checklist: Normality – look at histogram of residuals Heteroscedasticity – plot residuals with each x variable Autocorrelation – if data has a natural order, plot residuals in order and check for a pattern

The Grand Checklist• Linearity: scatter plot, common sense, and knowing your problem, transform including interactions if useful• t-statistics: are the coefficients significantly different from zero? Look at width of confidence intervals • F-tests for subsets, equality of coefficients tests for subsets, • R2: is it reasonably high in the context? • Influential observations, outliers in predictor space, dependent variable space• Normality: plot histogram of the residuals•Studentized residuals• Heteroscedasticity : plot residuals with each x variable, transform if necessary, Box-Cox transformations • Autocorrelation:”time series plot” • Multicollinearity : compute correlations of the x variables, do signs of coefficients agree with intuition?• Principal Components • Missing Values