Model Econometrics 计量经济学 Parameters of interest代写

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Model Econometrics 计量经济学 Parameters of interest代写

Week 8-‐ Chapter 13 – Recap Bivariate Regression & Multiple Regression

ECMT 1020: Introduction to Econometrics

Lecture 8

Instructor: Yi Sun

Contact: yi.sun@sydney.edu.au

School of Economics

The University of Sydney

Week 8

Week 8-‐ Chapter 13 – Recap Bivariate Regression & Multiple Regression

Contact Information

Ø Instructor W7-W13: Yi Sun

o Email：yi.sun@sydney.edu.au

o Office: Room 483, Merewether Building (H04)

o Office Hours: Tuesday 15:30-17:30

Friday 15:00-17:00 or by appointment

• Some Rules:

o You should contact me by email.

o Use your USyd email - identify yourself with your name and SID

o Any questions regarding the tutorial program including administrative matters

regarding tutorial allocation should be directed to your tutor

Week 8-‐ Chapter 13 – Recap Bivariate Regression & Multiple Regression

In general, we are interested the following aspects of a model:

Ø Model and parameters of interest

o What is the model?

o What are parameters we are interested in?

Ø Estimation

o If we have a data set, how do we estimate the parameters we want?

Ø Properties of the estimator (depends on assumptions)

o Are the estimated values of the parameters informative?

Ø Inference (hypothesis testing, confidence interval and so on)

o What can we say about the true value of the parameters?

Ø Interpretation

o What do our results mean in the specific problem we work on?

Week 8-‐ Chapter 13 – Recap Bivariate Regression & Multiple Regression

Ø Prediction

o How to make predictions based on our model?

Ø Evaluation and Comparison

o Is this a good model that fits the data well?

Ø Important special cases

o Data Transformations

Week 8-‐ Chapter 13 – Recap Bivariate Regression & Multiple Regression

For Bivariate Regression

Ø Model:

! = ! ! + ! ! ! + !

Ø Parameters of interest:

! ! and ! !

(Strictly speaking, for this to be a model, we also need “u is an error term”.)

Week 8-‐ Chapter 13 – Recap Bivariate Regression & Multiple Regression

Ø Estimation: Ordinary Least Squares (OLS)

• Given the data: ! ! ,! ! ,! = 1,2,…,!, we look for values ! ! and ! ! that

minimizes:

! ! − ! ! − ! ! ! !

!!!

• The solutions are:

! ! = ! − ! ! !

! ! =

(! ! − !)(! ! − !)

!!!

(! ! − !) !

!!!

= !

! !

• Based on the estimators, we define:

o Fitted line, residual, TSS, ExpSS, RSS and so on.

Week 8-‐ Chapter 13 – Recap Bivariate Regression & Multiple Regression

Intuition of the regression line:

Week 8-‐ Chapter 13 – Recap Bivariate Regression & Multiple Regression

Ø Assumptions:

• Population Assumptions:

1- The population model is

! ! = ! ! + ! ! ! ! + ! ! for all i

2- The error has zero mean conditional on the regressor

! ! ! ! ! = 0 for all i

3- The error has constant variance conditional on regressor

!"# ! ! ! ! = ! !

! for all i

4- The errors for different observations are statistically independent

! ! is independent of ! ! for all ! ≠ !

Week 8-‐ Chapter 13 – Recap Bivariate Regression & Multiple Regression

• Extra assumptions we need on Data:

There are some variations in the regressor (so that (! !

−!) ! ≠ 0). And we

have at least 3 observations.

Comments on assumptions

• Assumptions on data are very mild. They just ensure that

o ! ! can be computed from the data using our formula.

• Population assumptions are restrictive, but can be relaxed

o It’s an important topic in more advanced econometrics class. And it is

fascinating!

Week 8-‐ Chapter 13 – Recap Bivariate Regression & Multiple Regression

Intuition of the population assumptions:

• Assumption 1 – The relationship is linear in parameters.

• Assumption 2 – For any value of !, the error ! on average equals zero. This

implies that error term is uncorrelated with the regressor.

• Assumption 3 –Conditional on !, the variance of error term ! that does not vary

with the value of ! – Homoskedastic errors.

• Assumption 4 – The value taken by the error u for one observation is independent

of the value of the error for other observations.

Week 8-‐ Chapter 13 – Recap Bivariate Regression & Multiple Regression

• Assumptions 1 & 2 are the crucial assumptions that ensure that

! ! ! ! ! = ! ! + ! ! ! !

• Assumptions 3& 4 are additional assumptions that are used in determining

the precision and distribution of the estimates of ! ! !"# ! ! .

Week 8-‐ Chapter 13 – Recap Bivariate Regression & Multiple Regression

Ø Properties:

• ! ! !"# ! ! are unbiased and consistent

• We know how to compute !"(! ! ) and !"(! ! ) using our sample.

o If Assumption 1-4 hold, we have formulas for them

o In general, they are provided in the stata output.

• (! ! − ! ! )/!"(! ! ) has a distribution that we know how to approximate.

This property make it possible to carry out inference on ! ! . Same for ! ! .

• If Assumption 1-4 hold, OLS is BLUE. In additional, if the errors are normal,

OLS is BUE.

Week 8-‐ Chapter 13 – Recap Bivariate Regression & Multiple Regression

Ø Inference:

What can we learn about ! ! and ! ! given the model and the data?

Week 8-‐ Chapter 13 – Recap Bivariate Regression & Multiple Regression

• Define

! =

! ! − ! !

!" ! !

• Under Assumptions 1-4

o We approximate the distribution of T by a ! distributed with (! − 2) degrees

of freedom.

o This approximation is exact if errors are normal or ! → ∞.

• We use this result for our statistical inference

• Similar for ! !

Week 8-‐ Chapter 13 – Recap Bivariate Regression & Multiple Regression

Ø Interpretation

• ! ! : intercept (predicted value of Y when X equals 0)

• ! ! : slope, or marginal effect (predicted ∆! for a one unit increase in X)

Ø Prediction

• Conditional mean of ! at a given value of !:

! !" = ! ! + ! ! !

• Point prediction of ! at a given value of !:

! ! = ! ! + ! ! !

Values for these two predicted quantities at the same ! are the same, but standard

error for these two predicted values are different.

Week 8-‐ Chapter 13 – Recap Bivariate Regression & Multiple Regression

Ø Evaluation and Comparison

o We can use ! ! to measure the fit of a model when an intercept is included in

the model.

o ! ! only captures linear association between the dependent variable and the

regressor.

o ! ! can be used to compare bivariate (an intercept and one regressor)

regression models with same dependent variable.

o Low ! ! does not mean the regression analysis is useless.

Week 8-‐ Chapter 13 – Recap Bivariate Regression & Multiple Regression

Ø Important special cases: Data Transformation

o Indicator variable:

§ Interpretation: Difference in means across different groups in data.

o Natural logarithms:

§ Interpretation: proportionate changes, elasticity and semi-elasticity.

§ Prediction: retransformation bias (if dependent variable is transformed).

Week 8-‐ Chapter 13 – Recap Bivariate Regression & Multiple Regression

Multivariate Data

• We have seen how to analyse univariate data and bivariate data

• Now it is time to move on to working with more than two variables

• Most of what we do in economics uses more than two variables, even if the

question of interest is the relationship between X and Y

• Why? Because we're never in a controlled environment, there are lots of things

other than X and Y moving around, which may affect our analysis.

Week 8-‐ Chapter 13 – Recap Bivariate Regression & Multiple Regression

• For example, suppose we have the following true relation

!"#$ℎ !"#$ !" !"# !""#$%&'( = 1 + 4 ∗ !"##$ − 3 ∗ !"#$%&

!"#$%& = 2 ∗ !"##$ + !"#$%& !""#"

• If we are interested in the relation between death rate in car accidents and speed,

and we estimate their relation using a bivariate regression model, we will get:

!"#$ℎ !"#$ !" !"# !""#$%&'(

= 1 + 4 ∗ !"##$ − 3 ∗ 2 ∗ !"##$ + !"#$%& !""#"

= 1 − 2 ∗ !"##$ − 3 ∗ !"#$%& !""#"

• In this case, the coefficient on speed will be biased, and inconsistent.

• Moreover, the coefficient even has the wrong sign! We will conclude that as

speed goes up, death rate goes down. This is wrong!

Week 8-‐ Chapter 13 – Recap Bivariate Regression & Multiple Regression

• How to avoid it? Include both speed and safety in our analysis!

• In general, if a variable ! (safety) is correlated with both dependent variable

(death rate) and regressor(s) (speed), then we should include it in our analysis.

• If we omit a variable ! like this, OLS estimators will typically be biased and

inconsistent.

• This problem is called omitted variable bias.

• It can happen in multiple regressions too.

Week 8-‐ Chapter 13 – Recap Bivariate Regression & Multiple Regression

The general plan for studying multivariate data:

• Data description

• Model: Multivariate regression

o Model and parameters of interest

o Estimation

o Assumptions and properties

o Inference

o Interpretation

o Prediction

o Evaluation and Comparison

o Important Special Cases

Week 8-‐ Chapter 13 – Recap Bivariate Regression & Multiple Regression

Graphing Multivariate Data

• With three variables, you can do a three-way scatter-plot

15

Model Econometrics 计量经济学 Parameters of interest代写

educ

exper

wage

Week 8-‐ Chapter 13 – Recap Bivariate Regression & Multiple Regression

• With additional variables, you have to start getting creative (3-D surface with

color, animation to show a time dimension, etc.)

• An alternative is to produce a scatterplot for every pairing of variables

Week 8-‐ Chapter 13 – Recap Bivariate Regression & Multiple Regression

wage

educ

exper

0 10 20 30

0 10 20

0 20 40 60

Week 8-‐ Chapter 13 – Recap Bivariate Regression & Multiple Regression

• Graphs aren't going to get us too far with multivariate data

• Instead, the most common approach is to use a multivariate regression model

• This approach assumes that we have one dependent variable of interest (y)

• Now, we have several independent variables (x) and need some new notations

• We now have k random variables:

o Y : dependent variable, outcome, left-hand-side (LHS) variable

o ! ! ,! ! … ! ! : covariates, explanatory variables, independent variables, right-

hand-side (RHS) variables, regressors

Week 8-‐ Chapter 13 – Recap Bivariate Regression & Multiple Regression

• Our model is now:

! = ! ! + ! ! ! ! + ! ! ! ! …+ ! ! ! ! + !

• Parameters of interest: ! ! ,! ! ,…,! !

Week 8-‐ Chapter 13 – Recap Bivariate Regression & Multiple Regression

• Estimation: similar to the Bivariate Regression Model, we want to find a “line”

that fit the data “best”.

• In the multivariate case, a “line” is represented as:

! = ! ! + ! ! ! ! + ! ! ! ! + ⋯+ ! ! ! !

• For individual i, we observe the realization of (! ! ,! ! ,…,! ! ), which is

(! !! ,! !! ,…,! !" ).

• And ! ! (the value of ! for individual i predicted by this line) is:

! ! = ! ! + ! ! ! !! + ! ! ! !! + ⋯+ ! !" ! !"

Week 8-‐ Chapter 13 – Recap Bivariate Regression & Multiple Regression

• ! ! ,! ! ,…,! ! can take any values, so there are many lines.

• Which line should we choose? One that fits the data “best”.

• What do we mean by “best” then?

• The Ordinary Least Squares (OLS) estimator for ! ! ,! ! ,…,! ! is the values of

! ! ,! ! ,…,! ! that solves

!"# ! ! ,! ! ,…,! !

! ! − ! !

!!!

or equivalently,

!"# ! ! ,! ! ,…,! !

! ! − ! ! − ! ! ! !! − ⋯− ! ! ! !"

! !

!!!

Week 8-‐ Chapter 13 – Recap Bivariate Regression & Multiple Regression

• The main idea is the same as in the bivariate regression model.

• The difference now is that we have more variables, and that the minimization is

done by choosing the values of k different coefficients

• To solve this, we would take the derivative with respect to each ! ! and set it equal

to zero.

• This would give us k different equations to solve for k different unknowns

∑ !!!

! ! = 0

∑ !!!

! !" ! ! = 0, ! = 2,…,!

Intuition:

o The residuals sum to 0.

o Each regressor is orthogonal (uncorrelated) to the residual.

Week 8-‐ Chapter 13 – Recap Bivariate Regression & Multiple Regression

• Under mild conditions on the data (we need sufficient variations in the data on all

the !), this problem has a unique solution.

• The solution gives us a way to calculate each ! ! as a function of our data

• The coefficients aren't hard to derive if you know a little matrix algebra, but it’s

hard to write it out without using matrix notation.

• Interpretation for ! ! : the partial effect on the predicted value of y when ! !

changes by one unit, holding ! ! ,…,! ! constant. This is also called the effect of ! !

ceteris paribus.

• In general, the value of ! ! is different from the slope in the bivariate regression of

y on ! ! (and an intercept).

Week 8-‐ Chapter 13 – Recap Bivariate Regression & Multiple Regression

• We'll just use STATA's regression option to calculate them.

• The same STATA command for ! = ! ! + ! ! ! ! + ! ! ! ! + !

reg y x 2 x 3

• The regression output will contain coefficients, standard errors, t-stats, p-values

for all variables (Also ANOVA)

Week 8-‐ Chapter 13 – Recap Bivariate Regression & Multiple Regression

• Earnings Example :

!"#$%$&' = ! ! + ! ! ! + ! ! !"#!$ + !

S= years of Schooling ,

EXP= years of labour market experience

• Say we are primarily interested in the effect of S on earnings (! ! ):

ð the model explicitly controls for the effect of experience

ð we measure the effect of S on wages holding experience fixed

ð still need to make assumptions about how u is related to the explanatory

variables

ð in the simple regression model, exper was in u so we needed to assume exper

and S were independent, which is very unlikely

• Note ! ! : measures the cet. par. effect of experience on earnings, which may also

be of interest.

Week 8-‐ Chapter 13 – Recap Bivariate Regression & Multiple Regression

• Earnings Example :

!"#$%$&' = ! ! + ! ! ! + ! ! !"#!$ + !

S= years of Schooling,

EXP= years of labour market experience

reg EARNINGS S EXP

Source | SS df MS Number of obs = 540

-------------+------------------------------ F( 2, 537) = 67.54

Model | 22513.6473 2 11256.8237 Prob > F = 0.0000

Residual | 89496.5838 537 166.660305 R-squared = 0.2010

-------------+------------------------------ Adj R-squared = 0.1980

Total | 112010.231 539 207.811189 Root MSE = 12.91

------------------------------------------------------------------------------

EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval]

-------------+----------------------------------------------------------------

S | 2.678125 .2336497 11.46 0.000 2.219146 3.137105

EXP | .5624326 .1285136 4.38 0.000 .3099816 .8148837

_cons | -26.48501 4.27251 -6.20 0.000 -34.87789 -18.09213

------------------------------------------------------------------------------

EXP S INGS N EAR 56 . 0 68 . 2 49 . 26

+ + + + − − = =

Week 8-‐ Chapter 13 – Recap Bivariate Regression & Multiple Regression

reg EARNINGS S EXP

Source | SS df MS Number of obs = 540

-------------+------------------------------ F( 2, 537) = 67.54

Model | 22513.6473 2 11256.8237 Prob > F = 0.0000

Residual | 89496.5838 537 166.660305 R-squared = 0.2010

-------------+------------------------------ Adj R-squared = 0.1980

Total | 112010.231 539 207.811189 Root MSE = 12.91

------------------------------------------------------------------------------

EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval]

-------------+----------------------------------------------------------------

S | 2.678125 .2336497 11.46 0.000 2.219146 3.137105

EXP | .5624326 .1285136 4.38 0.000 .3099816 .8148837

_cons | -26.48501 4.27251 -6.20 0.000 -34.87789 -18.09213

------------------------------------------------------------------------------

!"#!!"#$ = −!".!" + !.!"! + !.!"!"#

(4.27) (0.23) (0.13)

Week 8-‐ Chapter 13 – Recap Bivariate Regression & Multiple Regression

reg EARNINGS S EXP

Source | SS df MS Number of obs = 540

-------------+------------------------------ F( 2, 537) = 67.54

Model | 22513.6473 2 11256.8237 Prob > F = 0.0000

Residual | 89496.5838 537 166.660305 R-squared = 0.2010

-------------+------------------------------ Adj R-squared = 0.1980

Total | 112010.231 539 207.811189 Root MSE = 12.91

------------------------------------------------------------------------------

EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval]

-------------+----------------------------------------------------------------

S | 2.678125 .2336497 11.46 0.000 2.219146 3.137105

EXP | .5624326 .1285136 4.38 0.000 .3099816 .8148837

_cons | -26.48501 4.27251 -6.20 0.000 -34.87789 -18.09213

------------------------------------------------------------------------------

!"#!!"#$ = −!".!" + !.!"! + !.!"!"#

(4.27) (0.23) (0.13)

èthe coefficient on S means that, holding experience fixed, an extra year of education

is predicted to increase earnings by 2.68 $

Week 8-‐ Chapter 13 – Recap Bivariate Regression & Multiple Regression

reg EARNINGS S EXP

Source | SS df MS Number of obs = 540

-------------+------------------------------ F( 2, 537) = 67.54

Model | 22513.6473 2 11256.8237 Prob > F = 0.0000

Residual | 89496.5838 537 166.660305 R-squared = 0.2010

-------------+------------------------------ Adj R-squared = 0.1980

Total | 112010.231 539 207.811189 Root MSE = 12.91

------------------------------------------------------------------------------

EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval]

-------------+----------------------------------------------------------------

S | 2.678125 .2336497 11.46 0.000 2.219146 3.137105

EXP | .5624326 .1285136 4.38 0.000 .3099816 .8148837

_cons | -26.48501 4.27251 -6.20 0.000 -34.87789 -18.09213

------------------------------------------------------------------------------

èthe coefficient on S means that, holding experience fixed, an extra year of education

is predicted to increase earnings by 2.68 $

ð equivalently, if we have 2 people with the same experience, the coefficient on

education reports the difference in their predicted earnings when their

education differs by 1 year

Week 8-‐ Chapter 13 – Recap Bivariate Regression & Multiple Regression

• Let’s introduce more explanatory variables – also a slight change in the

dependent variable.

!"!(!"#$%!"#) = ! ! + ! ! ! + ! ! !"# + ! ! !"#$%" + !

Where : S=years of Schooling ,EXP= years of labour market

experience, TENURE= years with current employer

. reg LogEARNINGS S EXP TENURE

Source | SS df MS Number of obs = 540

-------------+------------------------------ F( 3, 536) = 71.66

Model | 53.4473557 3 17.8157852 Prob > F = 0.0000

Residual | 133.260288 536 .248619939 R-squared = 0.2863

-------------+------------------------------ Adj R-squared = 0.2823

Total | 186.707643 539 .34639637 Root MSE = .49862

------------------------------------------------------------------------------

LogEARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval]

-------------+----------------------------------------------------------------

S | .1215696 .0090472 13.44 0.000 .1037972 .1393419

EXP | .0290606 .0053196 5.46 0.000 .0186108 .0395105

TENURE | .0112728 .0035815 3.15 0.002 .0042374 .0183082

_cons | .5594394 .1657859 3.37 0.001 .2337697 .8851092

------------------------------------------------------------------------------

Week 8-‐ Chapter 13 – Recap Bivariate Regression & Multiple Regression

!"#(!"!"#!"#) = 0.559 + 0.122! + 0.029!"# + 0.011!"#$%"

èthe coefficient on S means that, holding experience and tenure fixed, an extra year

of education is predicted to increase log(earnings) by 0.122.

èapproximately 12.2% increase in earnings

ð Equivalently, if we have 2 people with the same experience and tenure, the

coefficient on education reports the proportional difference in their predicted

earnings when their education differs by 1 year.

Week 8-‐ Chapter 13 – Recap Bivariate Regression & Multiple Regression

Changing more than one independent variable simultaneously

• Sometimes we are interested in the effect of changing more than one

independent variable at the same time

ð this is straightforward using fitted regression line

ð for example, what is the estimated effect on earnings when an individual stays

at the same firm for another year, and hence both experience and tenure

increase by 1 year ?

The aggregate effect is:

∆!!"(!"!"#!"#) = 0.029∆!"# + 0.011∆!"#$%"

= 0.029 + 0.011

= 0.040 or approximately 4% in earnings

Week 8-‐ Chapter 13 – Recap Bivariate Regression & Multiple Regression

Fitted Values and Residuals.

• Once we have the OLS regression line, we can calculate the fitted or predicted

value for each observation:

! ! = ! ! + ! ! ! !! + ! ! ! !! …+ ! ! ! !"

• we just plug the values of the independent variables into the OLS regression

line to get the predicted values

• STATA after running OLS regression ( post estimation command)

predict yhat, xb /* “,xb” specifies we want fitted values , and the new

variable name is yhat */

Week 8-‐ Chapter 13 – Recap Bivariate Regression & Multiple Regression

The residual for observation i is

! ! = ! ! − ! !

The OLS fitted values and residuals have some important properties:

1. the same average of the residuals is 0

2. the sample covariance between each independent variable and the OLS residuals is

0. As a result, the sample covariance between the OLS fitted values and the residuals

is 0.

3. The point ! ! ,! ! ,! ! ,! ! … ! ! ,! is always on the regression line.

STATA command – post estimation command

predict residual, resid /* “,resid” specifies we want residuals , and the new

variable name is residual */

Week 8-‐ Chapter 13 – Recap Bivariate Regression & Multiple Regression

• What are the conditions we need on data to have a unique solution ! ! ,! ! ,…,! ! ?

• First, we need at least k observations.

• Second, we need to have adequate variation in the regressors

• What do we mean by adequate variation?

Week 8-‐ Chapter 13 – Recap Bivariate Regression & Multiple Regression

Perfectly Collinear Regressors: Example 1

• We want to study how earnings vary with age, education and experience.

• True relation:

!"#$%$&' = 10 + !"# + !"ℎ!!" + 2 ∗ !"#$%&$'($

• However, if everyone enters school at age 6 and starts working as soon as they

leave school, then

!"#$%&$'($ = !"# − !"ℎ!!" − 6

• Then we can write

!"#$%$&' = 10 + !"# + !"ℎ!!" + 2 ∗ !"# − !"ℎ!!" − 6

= −2 + 3 ∗ !"# − !"ℎ!!"

Week 8-‐ Chapter 13 – Recap Bivariate Regression & Multiple Regression

• We can even write:

!"#$%$&'

= 10 + !"# + !"ℎ!!" + 2 − ! + ! ∗ !"#$%&$'($

= 10 + !"# + !"ℎ!!" + 2 − ! ∗ !"#$%&$'($ + ! ∗ (!"# − !"ℎ!!" − 6)

= 10 − 6! + 1 + ! ∗ !"# + 1 − ! ∗ !"ℎ!!" + 2 − ! ∗ !"#$%&$'($

• If we regress earnings on age, school, experience with an intercept, there are

infinite many ways to write down the best fitting line.

• In this case, we don’t have a unique solution for OLS.

• This happens when there is a linear relationship among the regressors. Notice

!"#$%&$'($ = !"# − !"ℎ!!" − 6

holds for each observation in our data set.

• This is called perfectly collinear regressors.

Week 8-‐ Chapter 13 – Recap Bivariate Regression & Multiple Regression

Perfectly Collinear Regressors: Example 2

• We want to study how earnings vary with age, gender.

• True relation

!"#$%$&' = 10 + !"# + !

! = 1 !" !"#!$!#%&' ! !" !"#$

! = 0 !" !"#!$!#%&' ! !" !"#$%"

• Suppose in our sample everyone is male, so ! = 1 for all i.

• Then we can write

!"#$%$&' = 11 + !"#

= !"# + 11 ∗ !

= ! + !"# + 11 − ! ∗ !

Week 8-‐ Chapter 13 – Recap Bivariate Regression & Multiple Regression

• Again, if we regress earnings on age and indicator for male with an intercept,

there are infinite many ways to write down the best fitting line.

• Again, this happens when there is a linear relationship among the regressors.

! = 1 for every individual in our data set

• When this happens, Stata will drop some of the trouble making variables

automatically, and display regression output on the variables left.

• However, we need to understand what is going on, and what Stata is doing.

Week 8-‐ Chapter 13 – Recap Bivariate Regression & Multiple Regression

• In general, we have a problem if one (or more) of the regressors can be expressed

as an exact linear combination of the other regressors.

• When this happens, we don’t have a unique solution for ! ! ,! ! ,…,! ! .

• Stata will drop some of the trouble making variables automatically

• A related problem is multicollinearity. This happens when one (or more) of the

regressors is very close to equalling an exact linear combination of the other

regressors.

• When this happens, we still have a unique solution for ! ! ,! ! ,…,! ! , and Stata runs

fine. But the estimated coefficdients on the trouble making variables are imprecise

and unstable.

Week 8-‐ Chapter 13 – Recap Bivariate Regression & Multiple Regression

Multivariate Regression: Goodness of Fit

• We can use the same methods as before to measure how good the fit of the

regression line is:

o The standard error of the regression

o The R 2

• We also have another measure called the adjusted R 2

• All of these measures are reported in STATA regression output

Week 8-‐ Chapter 13 – Recap Bivariate Regression & Multiple Regression

Multivariate Regression: Goodness of Fit

• The standard error of the regression:

! ! =

! − !

! ! − ! !

!!!

• This measures the average squared deviation of each ! ! from its predicted value

• It will be smaller the better our fit is but its magnitude depends on the units in

which we measure y.

• Another name for ! ! is the root mean squared error (RMSE) of the residual.

• As regressors are added to a model ! ! may decrease or may increase.

Week 8-‐ Chapter 13 – Recap Bivariate Regression & Multiple Regression

Multivariate Regression: Goodness of Fit

• The ! ! :

(a) Total Sum of Squares è TSS = ! ! − !

! !

!!!

(b) Explained Sum of Squares è ExpSS = ! ! − !

! !

!!!

! !

!!!

!"" = !"#$$ + !""

ð Total variation in the ! ! is the sum of the variation in ! ! and ! !

Week 8-‐ Chapter 13 – Recap Bivariate Regression & Multiple Regression

• The R-square (! ! ) is defined as :

! ! =

!"#$$

!""

= 1 −

!""

ð ! ! will be between 0 and 1 if we have an intercept

ð ! ! will be between 0 and 1, the closer it is to 1 the better fit of the regression

line is.

ð The problem with ! ! is that it will automatically increase (or at least stay the

same) whenever we add more regressors

ð We would like a measure that takes into account the number of regressors we

use

ð For example, we might prefer a line that gives us an ! ! of .7 with only three

regressors to a line that gives us an ! ! of .71 but uses forty regressors

Week 8-‐ Chapter 13 – Recap Bivariate Regression & Multiple Regression

ð The adjusted R-square (! ! ) is defined as :

! ! = 1 −

! − 1

! − !

!""

= 1 −

! !

• The adjusted R 2 (! ! ) will no greater than 1, and will be closer to 1 the better

the fit is.

• The adjusted R 2 is less than R 2 due to a modest penalty for the number of

regressors in the model

• Adding a regressor will raise the adjusted R 2 if it lowers the error sum of

squares enough to offset the penalty for increasing k.

• The adjusted R 2 can be less than 0 even if we include an intercept.

• Be careful about using the adjusted R 2 to evaluate models.

Week 8-‐ Chapter 13 – Recap Bivariate Regression & Multiple Regression

(a) (b)

!"#$ = ! ! + ! ! ! + ! !"#$ = ! ! + ! ! ! + ! ! !"# + !

Number of obs = 540

F( 1, 538) = 112.15

Prob > F = 0.0000

Model Econometrics 计量经济学 Parameters of interest代写

R-squared = 0.1725

Adj R-squared = 0.1710

Root MSE = 13.126

Number of obs = 540

F( 2, 537) = 67.54

Prob > F = 0.0000

R-squared = 0.2010

Adj R-squared = 0.1980

Root MSE = 12.91

(c)

!"#$ = ! ! + ! ! ! + ! ! !"# + ! ! !"# + !

Number of obs = 540

F( 3, 537) = 46.27

Prob > F = 0.0000

R-squared = 0.2057

Adj R-squared = 0.2012

Root MSE = 12.88

Week 8-‐ Chapter 13 – Recap Bivariate Regression & Multiple Regression

Number of obs = 540

F( 3, 537) = 46.27

Prob > F = 0.0000

R-squared = 0.2057

Adj R-squared = 0.2012

Root MSE = 12.88

(d)!"#$ = ! ! + ! ! ! + ! ! !"# + ! ! !"# + ! ! !"#$% + !

Source | SS df MS Number of obs = 540

-------------+------------------------------ F( 4, 535) = 34.68

Model | 23064.3736 4 5766.0934 Prob > F = 0.0000

Residual | 88945.8575 535 166.253939 R-squared = 0.2059

-------------+------------------------------ Adj R-squared = 0.2000

Total | 112010.231 539 207.811189 Root MSE = 12.894

------------------------------------------------------------------------------

EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval]

-------------+----------------------------------------------------------------

S | 2.646137 .234041 11.31 0.000 2.186385 3.105889

EXP | .4735452 .1375807 3.44 0.001 .2032804 .7438099

TENURE | .1697577 .0935248 1.82 0.070 -.0139631 .3534785

URBAN | .4475683 1.161625 0.39 0.700 -1.834338 2.729475

_cons | -26.06358 4.362455 -5.97 0.000 -34.63322 -17.49394

------------------------------------------------------------------------------

Model Econometrics 计量经济学 Parameters of interest代写