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Multiple

Regression

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PADP 8130 - Week 2

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Slides directly adapted from Introductory Econometrics: A Modern Approach, 6th Edition (Wooldridge)

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Multiple Regression Analysis: Estimation

Definition of the multiple linear regression model:

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Dependent variable,

explained variable,

response variable,…

Independent variables,

explanatory variables,

regressors,…

Error term,

disturbance,

unobservables,…

Intercept

Slope parameters

“Explains variable in terms of variables ”

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Multiple Regression Analysis: Estimation

  • Motivation for multiple regression
    • Incorporate more explanatory factors into the model
    • Explicitly hold fixed other factors that otherwise would be in
    • Allow for more flexible functional forms
  • Example: Wage equation

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Hourly wage

Years of education

Years of labor market experience

All other factors…

Now measures effect of education explicitly holding experience fixed

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Multiple Regression Analysis: Example 2

  • Example: Average test scores and per student spending

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    • Per student spending is likely to be correlated with average family income at a given high school because of school financing
    • Omitting average family income in regression would lead to biased estimate of the effect of spending on average test scores
    • In a simple regression model, effect of per student spending would partly include the effect of family income on test scores

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Average standardized

test score of school

Other factors

Per student spending

at this school

Average family income

of students at this school

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Multiple Regression Analysis: Example 3

  • Example: Family income and family consumption

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    • Model has two explanatory variables: income and income squared
    • Consumption is explained as a quadratic function of income
    • One has to be very careful when interpreting the coefficients:

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Family consumption

Other factors

Family income

Family income squared

By how much does consumption

increase if income is increased

by one unit?

Depends on how much income is already there

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Multiple Regression Analysis: Example 4

  • Example: CEO salary, sales, and CEO tenure

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    • Model assumes a constant elasticity relationship between CEO salary and the sales of his or her firm
    • Model assumes a quadratic relationship between CEO salary and his or her tenure with the firm
  • Meaning of “linear” regression
    • The model has to be linear in the parameters (not in the variables)

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Log of CEO salary

Log sales

Quadratic function of CEO tenure with the firm

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Multiple Regression Analysis: Estimation

OLS Estimation of the multiple regression model

  • Random sample

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  • Regression residuals

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  • Minimize sum of squared residuals

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Minimization will be carried out by computer

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Multiple Regression Analysis: Estimation

  • Interpretation of the multiple regression model

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    • The multiple linear regression model manages to hold the values of other explanatory variables fixed even if, in reality, they are correlated with the explanatory variable under consideration
    • “Ceteris paribus”-interpretation
    • It has still to be assumed that unobserved factors do not change if the explanatory variables are changed

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By how much does the dependent variable change if the j-th independent variable is increased by one unit, holding all other independent variables and the error term constant

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Multiple Regression Analysis: Estimation

  • Example: Determinants of college GPA

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  • Interpretation
    • Holding ACT fixed, another point on high school grade point average is associated with another .453 points college grade point average
    • Or: If we compare two students with the same ACT, but the hsGPA of student A is one point higher, we predict student A to have a colGPA that is .453 higher than that of student B
    • Holding high school grade point average fixed, another 10 points on ACT are associated with less than one point on college GPA

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Grade point average at college

High school grade point average

Achievement test score

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Multiple Regression Analysis: Estimation

  • Properties of OLS on any sample of data
  • Fitted values and residuals

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  • Algebraic properties of OLS regression

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Fitted or predicted values

Residuals

Deviations from regression line sum up to zero

Covariance between deviations and regressors are zero

Sample averages of y and of the regressors lie on regression line

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Multiple Regression Analysis: Estimation

  • “Partialling out” interpretation of multiple regression
  • One can show that the estimated coefficient of an explanatory variable in a multiple regression can be obtained in two steps:
    • 1) Regress the explanatory variable on all other explanatory variables
    • 2) Regress on the residuals from this regression
  • Why does this procedure work?
    • The residuals from the first regression is the part of the explanatory variable that is uncorrelated with the other explanatory variables
    • The slope coefficient of the second regression therefore represents the isolated effect of the explanatory variable on the dep. variable

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Multiple Regression Analysis: Estimation

  • Goodness-of-Fit
  • Decomposition of total variation

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  • R-squared

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  • Alternative expression for R-squared

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Notice that R-squared can only increase if another explanatory variable is added to the regression

R-squared is equal to the squared correlation coefficient between the

actual and the predicted value of the dependent variable

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Multiple Regression Analysis: Estimation

  • Example: Explaining arrest records

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  • Interpretation:
    • If the proportion prior arrests increases by 0.5...

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    • If the months in prison increase from 0 to 12...

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    • If the quarters employed increase by 1...

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Number of times

arrested 1986

Proportion prior arrests

that led to conviction

Months in prison 1986

Quarters employed 1986

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Multiple Regression Analysis: Estimation

  • Example: Explaining arrest records

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  • Interpretation:
    • If the proportion prior arrests increases by 0.5, the predicted fall in arrests is 7.5 arrests per 100 men
    • If the months in prison increase from 0 to 12, the predicted fall in arrests is 0.408 arrests for a particular man
    • If the quarters employed increase by 1, the predicted fall in arrests is 10.4 arrests per 100 men

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Number of times

arrested 1986

Proportion prior arrests

that led to conviction

Months in prison 1986

Quarters employed 1986

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Multiple Regression Analysis: Estimation

  • Example: Explaining arrest records (cont.)
    • An additional explanatory variable is added:

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  • Interpretation:
    • Average prior sentence increases number of arrests (?)
    • Limited additional explanatory power as R-squared increases by little
  • General remark on R-squared
    • Even if R-squared is small (as in the given example), regression may still provide good estimates of ceteris paribus effects

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Average sentence in prior convictions

R-squared increases only slightly

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Multiple Regression Analysis: Estimation

  • Standard assumptions for the multiple regression model
  • Assumption MLR.1 (Linear in parameters)

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  • Assumption MLR.2 (Random sampling)

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In the population, the relationship between y and the explanatory variables is linear

The data is a random sample drawn from the population

Each data point therefore follows the population equation

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Multiple Regression Analysis: Estimation

  • Standard assumptions for the multiple regression model (cont.)
  • Assumption MLR.3 (No perfect collinearity)

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  • Remarks on MLR.3
    • The assumption only rules out perfect collinearity/correlation between explanatory variables; imperfect correlation is allowed
    • If an explanatory variable is a perfect linear combination of other explanatory variables it is superfluous and may be eliminated
    • Constant variables are also ruled out (collinear with intercept)

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“In the sample (and therefore in the population), none

of the independent variables is constant and there are

no exact linear relationships among the independent variables.”

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Multiple Regression Analysis: Estimation

  • Example for perfect collinearity: small sample

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  • Example for perfect collinearity: relationships between regressors

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In a small sample, avginc may accidentally be an exact multiple of expend; it will not be possible to disentangle their separate effects because there is exact covariation

Either shareA or shareB will have to be dropped from the regression because there is an exact linear relationship between them: shareA + shareB = 1

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Multiple Regression Analysis: Estimation

  • Standard assumptions for the multiple regression model (cont.)
  • Assumption MLR.4 (Zero conditional mean)

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    • In a multiple regression model, the zero conditional mean assumption is much more likely to hold because fewer things end up in the error
  • Example: Average test scores

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The value of the explanatory variables must contain no information about the mean of the unobserved factors

If avginc was not included in the regression, it would end up in the error term; it would then be hard to defend that expend is uncorrelated with the error

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Multiple Regression Analysis: Estimation

  • Discussion of the zero mean conditional assumption
    • Explanatory variables that are correlated with the error term are called endogenous; endogeneity is a violation of assumption MLR.4
    • Explanatory variables that are uncorrelated with the error term are called exogenous; MLR.4 holds if all explanat. var. are exogenous
    • Exogeneity is the key assumption for a causal interpretation of the regression, and for unbiasedness of the OLS estimators
  • Theorem 3.1 (Unbiasedness of OLS)

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    • Unbiasedness is an average property in repeated samples; in a given sample, the estimates may still be far away from the true values

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Multiple Regression�Analysis: Estimation

  • Including irrelevant variables in a regression model

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  • Omitting relevant variables: the simple case

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= 0 in the population

No problem because .

However, including irrelevant variables may increase sampling variance.

True model (contains x1 and x2)

Estimated model (x2 is omitted)

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Multiple Regression�Analysis: Estimation

  • Omitted variable bias

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  • Conclusion: All estimated coefficients will be biased

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If x1 and x2 are correlated, assume a linear regression relationship between them

If y is only regressed

on x1 this will be the estimated intercept

If y is only regressed

on x1, this will be the estimated slope on x1

error term

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Multiple Regression�Analysis: Estimation

  • Example: Omitting ability in a wage equation

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  • When is there no omitted variable bias?
    • If the omitted variable is irrelevant or uncorrelated

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Will both be positive

The return to education will be overestimated because . It will look

as if people with many years of education earn very high wages, but this is partly

due to the fact that people with more education are also more able on average.

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Multiple Regression�Analysis: Estimation

  • Omitted variable bias: more general cases

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    • No general statements possible about direction of bias
    • Analysis as in simple case if one regressor uncorrelated with others
  • Example: Omitting ability in a wage equation

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True model (contains x1, x2, and x3)

Estimated model (x3 is omitted)

If exper is approximately uncorrelated with educ and abil, then the direction

of the omitted variable bias can be as analyzed in the simple two variable case.

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Multiple Regression�Analysis: Estimation

  • Standard assumptions for the multiple regression model (cont.)
  • Assumption MLR.5 (Homoskedasticity)

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  • Example: Wage equation

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  • Short hand notation

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The value of the explanatory variables

must contain no information about the variance of the unobserved factors

This assumption may also be hard

to justify in many cases

with

All explanatory variables are collected in a random vector

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Multiple Regression�Analysis: Estimation

  • Theorem 3.2 (Sampling variances of the OLS slope estimators)

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Under assumptions MLR.1 – MLR.5:

Variance of the error term

Total sample variation in

explanatory variable xj:

R-squared from a regression of explanatory variable xj on all other independent variables

(including a constant)

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Multiple Regression�Analysis: Estimation

  • Components of OLS Variances:
  • 1) The error variance
    • A high error variance increases the sampling variance because there is more “noise” in the equation
    • A large error variance necessarily makes estimates imprecise
    • The error variance does not decrease with sample size
  • 2) The total sample variation in the explanatory variable
    • More sample variation leads to more precise estimates
    • Total sample variation automatically increases with the sample size
    • Increasing the sample size is thus a way to get more precise estimates

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Multiple Regression�Analysis: Estimation

  • 3) Linear relationships among the independent variables

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    • Sampling variance of will be the higher the better explanatory variable can be linearly explained by other independent variables
    • The problem of almost linearly dependent explanatory variables is called multicollinearity (i.e. for some )

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Regress on all other independent variables (including a constant)

The R-squared of this regression will be the higher

the better xj can be linearly explained by the other independent variables

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Multiple Regression�Analysis: Estimation

  • An example for multicollinearity

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Average standardized

test score of school

Expenditures

for teachers

Expenditures for in-

structional materials

Other ex-

penditures

The different expenditure categories will be strongly correlated because if a school has a lot of resources it will spend a lot on everything.

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It will be hard to estimate the differential effects of different expenditure categories because all expenditures are either high or low. For precise estimates of the differential effects, one would need information about situations where expenditure categories change differentially.

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As a consequence, sampling variance of the estimated effects will be large.

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Multiple Regression�Analysis: Estimation

  • Discussion of the multicollinearity problem
    • In the above example, it would probably be better to lump all expenditure categories together because effects cannot be disentangled
    • In other cases, dropping some independent variables may reduce multicollinearity (but this may lead to omitted variable bias)

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Multiple Regression�Analysis: Estimation

    • Only the sampling variance of the variables involved in multicollinearity will be inflated; the estimates of other effects may be very precise
    • Note that multicollinearity is not a violation of MLR.3 in the strict sense
    • Multicollinearity may be detected through “variance inflation factors”

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As an (arbitrary) rule of thumb, the variance

inflation factor should not be larger than 10

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Multiple Regression�Analysis: Estimation

  • Variances in misspecified models
    • The choice of whether to include a particular variable in a regression can be made by analyzing the tradeoff between bias and variance

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    • It might be the case that the likely omitted variable bias in the misspecified model 2 is overcompensated by a smaller variance

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True population model

Estimated model 1

Estimated model 2

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Multiple Regression�Analysis: Estimation

  • Variances in misspecified models (cont.)

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  • Case 1:

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  • Case 2:

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Conditional on x1 and x2, the variance in model 2 is always smaller than that in model 1

Conclusion: Do not include irrelevant regressors

Trade off bias and variance; Caution: bias will not vanish even in large samples

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Multiple Regression�Analysis: Estimation

  • Estimating the error variance

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  • Theorem 3.3 (Unbiased estimator of the error variance)

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An unbiased estimate of the error variance can be obtained by subtracting the number of estimated regression coefficients from the number of observations. The number of observations minus the number of estimated parameters is also called the degrees of freedom. The n estimated squared residuals in the sum are not completely independent but related through the k+1 equations that define the first order conditions of the minimization problem.

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Multiple Regression�Analysis: Estimation

  • Estimation of the sampling variances of the OLS estimators

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  • Note that these formulas are only valid under assumptions MLR.1-MLR.5 (in particular, there has to be homoskedasticity)

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The true sampling

variation of the

estimated

The estimated samp-

ling variation of the

estimated

Plug in for the unknown

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Multiple Regression�Analysis: Estimation

  • Efficiency of OLS: The Gauss-Markov Theorem
    • Under assumptions MLR.1 - MLR.5, OLS is unbiased
    • However, under these assumptions there may be many other estimators that are unbiased
    • Which one is the unbiased estimator with the smallest variance?
    • In order to answer this question one usually limits oneself to linear estimators, i.e. estimators linear in the dependent variable

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May be an arbitrary function of the sample values of all the explanatory variables; the OLS estimator

can be shown to be of this form

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Multiple Regression�Analysis: Estimation

  • Theorem 3.4 (Gauss-Markov Theorem)
    • Under assumptions MLR.1 - MLR.5, the OLS estimators are the best linear unbiased estimators (BLUEs) of the regression coefficients, i.e.

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  • OLS is only the best estimator if MLR.1 – MLR.5 hold; if there is heteroskedasticity for example, there are better estimators.

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for all

for which .