econ 642, wednesday march 26, class 2model and least squares principle interpretation of...

Econ 642, Monday March 24, class 1

Econ 642, Wednesday March 26, class 2

Robert de Jong1

1Department of EconomicsOhio State University

Robert de Jong Econ 642, Wednesday March 26, class 2


Model and Least squares principleInterpretation of coefficientsSoftware, taking logarithms, and R2

Inference and GRE exampleModel assumptions

Outline

1 Econ 642, Monday March 24, class 1Model and Least squares principleInterpretation of coefficientsSoftware, taking logarithms, and R2






Simple regression: estimates modelyi = β0 + β1xi + ui

Multiple regression: estimates modelyi = β0 + β1xi1 + . . . + βkxik + ui

Interpretation of βj :

βj is the amount with which E(yi |xi1, . . . , xik ) increases if xij

goes up by one unit, keeping all other variables constant





Least Squares principle - simple regressionMinimizing

n∑

i=1

(yi − (β0 + β1xi))2

over all possible values of β0 and β1 gives

β1 =n ·

∑ni=1 xiyi −

∑ni=1 xi ·

∑ni=1 yi

n ·∑n

i=1 x2i − (

∑ni=1 xi)2

andβ0 = y − β1x .

Note y = n−1 ∑ni=1 yi , the average of the yi

The mathematical calculation requires being able to find theminimum of a function of two variables using differentiation





Least squares principle - multiple regressionminimize

n∑

i=1

(yi − (β0 + β1xi1 + . . . + βk xik ))2

This problem can be solved using matrix algebra





Outline







Interpretation of coefficients

We obtain the regression line

y = β0 + β1x

yi : demand for housing of individual # i, in dollars annuallyxi : income in dollars annually

β1 is how many dollars an individual is predicted to spend onhousing when income increases by $1

β0 is how many dollars an individual is predicted to spend onhousing when income equals $0





Example:

yi wage of individual ixi1 is nr. of years of educationxi2 is nr. of years of work experiencexi3 indicates male/female

Model:yi = β0 + β1xi1 + β2xi2 + β3xi3 + ui

β1 is the effect on wage of a 1 unit increase in nr. of years ofeducation, keeping nr. of years of work experience and genderconstant

β3 is the effect on wage of a 1 unit increase in the male/femalevariable, keeping nr. of years of work experience and nr. ofyears of education constant





Interesting question/hypothesis in the previous regression:β3 = 0 ?

Often the interesting questions are answered by multivariateregression and not by simple regression





“Obvious" commenty = β0 + β1x

If β1 ≈ 0, then xi does not “influence" yi

Earlier example: if β1 ≈ 0, then expenditure on housing is not“influenced" by one’s income

Often, important questions correspond to a coefficient of 0

Does being a woman negatively impact earnings potential?

Is there a correlation between percentage of foreigners in aneighborhood and the crime rate?

Is there a relationship between number of cans of sodasold in a stadium and temperature?





Minimizingn∑

i=1

(yi − (β0 + β1xi))2

over all possible values of β0 and β1 gives

β1 =n ·

∑ni=1 xiyi −

∑ni=1 xi ·

∑ni=1 yi

n ·∑n

i=1 x2i − (

∑ni=1 xi)2

andβ0 = y − β1x

Note y = n−1 ∑ni=1 yi , the average of the yi

The mathematical calculation requires being able to find theminimum of a function of two variables using differentiation





Outline







Computer software: Eviews will calculate the regressioncoefficients for us

Other packages that can do this:

Excel, or other spreadsheet program

SAS, SPSS

Stata, Eviews





Creating logarithms of variables

Remember: log(x) is increasing, log(1) = 0; so if we find apositive slope, this still suggests a positive correlation betweeny and x

Often in econometrics, we use logarithms of y and x instead ofy and x themselves

We call this a double-logarithmic specification instead of aregression in levels

Reasons:1 Interpretation: elasticity2 It works





Q: How much does the demand q for pizza go up if income iincreases by 1%?

A: by the income elasticity of the demand for pizza

Mathematically:∆q/q∆i/i

Now∆q/q∆i/i

≈dq/qdi/i

anddlog(q)/dq = 1/q, so dlog(q) ≈ dq/q

Conclusion:∆q/q∆i/i

≈d log(q)

d log(i)Robert de Jong Econ 642, Wednesday March 26, class 2




R2, the coefficient of determination

A measure between 0 and 1 of how well the data fit the model

0: poor fit; 1: perfect fit

used for:1 evaluating the quality of a regression2 comparing models with different with different data sets

and different functional form





R2 is the fraction of the variation in yi that is explained by themodel

Values you can expect for R2: totally silly guidelines

1 Time series: often “high" values: 0.8 - 0.99 range2 Cross-sections: often values in the 0.1 - 0.4 range3 Panel data: often in the 0.3 - 0.7 range





What R2 do we expect for:1 Regression of national consumption on national income2 Regression of cigarettes smoked on income3 Wage on nr. of years of education and and various other

regressors





Issues with R2:

1. Some regressions with low R2 can still be interesting(smoking on income) while some regressions with high R2 canbe uninteresting (macro regression, e.g. exports on nationalincome)

2. Adding variables to a regression will always make R2 go up(mathematical necessity)





Outline







Importance of inference

How do we obtain statistical proof that a coefficient equalszero ?

(or any other interesting value)

Examples:

yi apgar score; regressor incomeQuestion: does income affect apgar score?

yi wage; regressors: nr. of years of education, type ofprofession, male/female, etc.Question: does gender have an impact on compensation?





Econometrician’s mindsets

We have observed data: the yi and xi .

We can calculate the regression line: calculate β0 and β1.

The data are generated as follows:1 Start with xi

2 We move to the value suggested by the true regressionline: β0 + β1xi (Note: no “hats" on the betas here!

3 a random error ui is added:yi = β0 + β1xi + ui





Situation:1 We have true but unknown coefficients β0 and β1

2 We try to approximate these coefficients using β0 and β1

Example:yi : GPA in collegexi : GPA in high school

Question: what happens if xi has no impact on yi? (this is astatistical hypothesis)





Answer: if xi has no impact on yi , then β1 = 0

NOTE: if β1 = 0, then β1 will be positive or negative, but notexactly zero

However β1 will be close to 0 in statistical sense if β1 = 0

Question: what does this mean: β1 close to 0?

Testing the hypothesis β1 = 0: 1. t-values2. standard errors3. p-values





GRE scores: are GRE and SAT biased against women andethnic groups?

GRE i = 172.4+ 39.7 Gi+ 78.9 GPAi

(10.9) (10.4)

+0.203 SATM i+ 0.110 SATV i

(0.071) (0.058)

where

GRE i = score of i th student on test

Gi = 1 if student is male, 0 otherwise

GPAi = GPA in economics classas

SATM i = score on SAT-mathematical

SATV i = score on SAT-verbal





Things to note on previous transparency1 Dummy variable: a variable that takes on the values 0 and

1 only (male/female dummy)

This means that if Gi = 1 - i.e. for a man - we predict ahigher GRE score than if Gi = 0 (i.e. a woman)

2 Between parentheses: standard errors (we get those fromEviews output too!)

Standard errors measure the statistical uncertainty in thecoefficient

cf. margin of error in the Bush-Kerry election

95% confidence interval (likely values for the coefficient):

coefficient plus or minus 1.96 times the standard errorRobert de Jong Econ 642, Wednesday March 26, class 2




Example: estimated GRE equation

coefficient for Gi (gender: 1 male, 0 female): 39.7

suggests 39.7 extra GRE points for males

standard error: 10.9

Note: 39.7 is more than 1.96 standard errors away from 0

Conclusion: the positive coefficient for Gi is statistically“remote" from 0

Conclusion: we have statistical evidence that men score higheron the GRE





Revisit GRE example:

GRE i = 172.4+ 39.7 Gi+ 78.9 GPAi

(10.9) (10.4)

+0.203 SATM i+ 0.110 SATV i

(0.071) (0.058)

where

GRE i = score of i th student on test

Gi = 1 if student is male, 0 otherwise

GPAi = GPA in economics classes

SATM i = score on SAT-mathematical

SATV i = score on SAT-verbal





t-value: the number of standard errors that the coefficient isaway from zero

Therefore,

t-value =coefficient

standard error

1 t-values are the third column in Eviews output2 sometimes t-values instead of standard errors between

parentheses under coefficients3 coefficient more than 1.96 standard errors away from zero

⇐⇒ t-value exceeds 1.96





Outline







Theory of regression:

Model assumptions

⇒

Coefficients are normally distributed

⇒

Considering 1.96 standard errors is correct practice





1.96 standard error procedure corresponds to test at 95%confidence level

If we want to test at the 99% confidence level, we need 2.465standard errors

If absolute value of the t-value > 1.96, then we reject the nullhypothesis that the coefficient equals zero





Looking at 1.96 standard errors is only valid practice if themodel assumptions are correct

Conclusion:

We need to understand the model assumptions

Model assumptions are of a probabilistic nature





The model assumptions

Model assumptions: statistical and mathematical assumptionsthat are needed to make our “two standard errors” procedurework





Model assumptions1 y = β0 + β1x1 + β2x2 + . . . + β3x3 + u, where β0, β1, . . . , βk

are the unknown parameters (constants) of interest, and uis an unobservable random error or random disturbanceterm.

2 (random sampling) We have a random sample of nobservations from the above linear population model.

3 (zero conditional mean) E(u|x1, . . . , xk ) = 0.4 (no perfect collinearity) In the sample (and therefore in the

population), none of the independent is constant, and thereare no exact linear relationships among the independentvariables.

5 (homoskedasticity) Var(u|x1, . . . , xn) = σ2.





1. The regression model is linear in the coefficients, is correctlyspecified, and has an additive error term.

This means:1 No “relevant” omitted variables2 No nonlinearity

Implication: Regression of unemployment on minimum wageinvalid if the impact of a $5 to $7 change in minimum wage isbigger than the impact of a $12 to $14 change

Q: what is: “relevant" ?





2. Random sampling assumption: no autocorrelation

Two cases:

A. Cross-section: it seems unlikely that individuals in across-section influence each other’s behavior

Conclusion: This assumption is not very strict for across-section

B. Time series: this assumption can be problematic

Example: National consumption vs. national income: both timeseries are pretty well predictable from past values





3. All explanatory variables are uncorrelated with the error term.

This rules out that yi impacts xi inappropriately

Endogeneity: xi cannot be assumed given for yi





Examples: is endogeneity an issue in a regression of:1 Soda sales on temperature?2 Pizza consumption on income?3 Alcohol use on index of marital happiness?4 GDP on index of corruption in third world nations?5 National consumption on national income?6 Wage on male/female dummy?





4. No explanatory variable is a perfect linear function of anyother explanatory variable(s).

This is called: no multicollinearity

Rules out, e.g.1 a regression of wage on a constant, “education years”, and

“education years" (identical regressors)2 a regression of wage on a constant, “education years”, and

“education months", if the education months variable is 12times education years

3 regression of wage on a constant, a “female" dummyvariable, and a “male” dummy variable





5. The error term has a constant variance.

This is called homoskedasticity; non-constant variance of theerror term is called heteroskedasticity

Classical situation where this can fail:Regression of housing expenditure on income


econ 642, wednesday march 26, class 2model and least squares principle interpretation of...

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