1 Copyright © 2007 Thomson Asia Pte. Ltd. All rights reserved. Multiple Regression Analysis:...

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Multiple Regression Analysis: Further Issues

y = 0 + 1x1 + 2x2 + . . . kxk + u

Redefining Variables

Changing the scale of the y variable will lead to a corresponding change in the scale of the coefficients and standard errors, so no change in the significance or interpretation

Changing the scale of one x variable will lead to a change in the scale of that coefficient and standard error, so no change in the significance or interpretation

1 1 22 1

ˆˆ ˆ,

ˆvar ,ˆ

ˆ ˆˆ, ,ˆ 1

/ ˆ ˆ,/ 1

i i i ii i

r urj j

x x y y r y

y ySSE

Rs SST y y

t seSST Rse

SSR SSR qF c se

SSR n k

SER SSR n k

数据测度的影响

统计量 y（由盎司改为磅，1 磅=16

盎司）

统计量 x（每天抽烟数由支改为包，1包 20支）

ˆj ˆ 16j ˆ

packs ˆ20 cigs

ˆjse ˆ 16jse ˆ

packsse ˆ20 cigsse

不变 ˆ

packst

不变

F 不变 F 不变 2R 不变 2R 不变

SSR 216SSR SSR 不变

SER 16SER

SER 不变

ˆ ˆ 16j jc se

ˆpacks

ˆ ˆ20 cigs cigsc se

因变量或自变量以对数形式出现，改变度量单位不会影响斜率系数，只会改变截距项。

1 1log log logi ic y c y

1 1log log logj jc x c x

Beta Coefficients

Occasional you’ll see reference to a “standardized coefficient” or “beta coefficient” which has a specific meaning

Idea is to replace y and each x variable with a standardized version – i.e. subtract mean and divide by standard deviation

Coefficient reflects standard deviation of y for a one standard deviation change in x

Beta Coefficients (cont)

1 1 2 2

ˆ ˆ ˆ...where denote the z-score of , is the z score of , and so on. Andˆ ˆˆ ˆb ( / ) for 1,..., are called beta coefficients.

z b z b z b z errorz y z

Example 6.1 :Effects of Pollution on Housing Prices

Functional Form

OLS can be used for relationships that are not strictly linear in x and y by using nonlinear functions of x and y – will still be linear in the parameters

Can take the natural log of x, y or both Can use quadratic forms of x Can use interactions of x variables

Interpretation of Log Models

ln(y) = 0 + 1ln(x) + u

1 is the elasticity of y with respect to x ln(y) = 0 + 1x + u

1 is approximately the percentage change in y given a 1 unit change in x

y = 0 + 1ln(x) + u

1 is approximately the change in y for a 100 percent change in x

Why use log models? Log models are invariant to the scale of the

variables since measuring percent changes They give a direct estimate of elasticity For models with y > 0, the conditional distribution

is often heteroskedastic or skewed, while ln(y) is much less so

The distribution of ln(y) is more narrow, limiting the effect of outliers.

it can not be used if a variable takes on zero or negative values.

It is more difficult to predict the original variable when using a dependent variable in log form.

Some Rules of Thumb

variables used in log form: Dollar amounts that must be positive Very large variables, such as population variables used in level form: Variables measured in years Variables that are a proportion or percent

0 1 1 2 2

ˆ ˆ ˆˆlog log,

ˆˆlog

ˆˆ% 100 exp 1

ˆˆˆ ˆlog 1 logˆ1 exp

ˆexp 1

y x xgive x we get

y y y x

Quadratic Models

For a model of the form y = 0 + 1x + 2x2 + u we can’t interpret 1 alone as measuring the change in y with respect to x, we need to take into account 2 as well, since

ˆ ˆˆ 2 , so

ˆ ˆ ˆ 2

ˆ0, ,

ˆ max min

ˆ ˆ2

when that isx

y reach its i or value

More on Quadratic Models

Suppose that the coefficient on x is negative and the coefficient on x2 is positive

Then y is decreasing in x at first, but will eventually turn around and be increasing in x

0ˆ and 0ˆ when as same the

is which ,ˆ2ˆat be will

point turning the0ˆ and 0ˆFor

How to describe decreasing effect

How to describe increasing effect

Interaction Terms

For a model of the form y = 0 + 1x1 + 2x2 + 3x1x2 + u we can’t interpret 1 alone as measuring the change in y with respect to x1, we need to take into account 3 as well, since

1 3 21

, so to summarize

the effect of on we typically

evaluate the above at

Example: Effects of Attendance on Final Exam Performance

atndrte 系数为负，是否意味着听课对期末考试分数具有负面影响？

b1 仅考虑了 priGPA=0 时的影响。 atndrte和 priGPA•atndrte 系数估计值 t 值不显著，是否意

味着两者对期末考试分数无影响？ F 检验的 p 值为 0.014.

Atndrte对 stndfnl 的偏效应：

其含义是：在 priGPA 的平均水平（ 2.59 ）上， atndrte 提高 10 个百分点，使 stndfnl 比期末考试平均分数高出 0.078 倍。

0.0067 0.0056 2.59 0.0078

ˆ ˆ 2.59,ˆ ˆ 2.59

0 1 6 6

ˆ 2.592.59

stndfnl atndrte priGPA atndrte uatndrte priGPA atndrte u

10.0078 0.0026 3t

Adjusted R-Squared

Recall that the R2 will always increase as more variables are added to the model

The adjusted R2 takes into account the number of variables in a model, and may decrease

1 1 1 1

SSR nR

SST nSSR n k

RSST n

R n n k

调整 R 方的作用：为在一个模型中另外增加自变量施加了惩罚。

随着一个新的自变量加入回归方程， SSR 下降，但回归中的自由度 df=n-k-1 也下降。因此，SSR/(n-k-1) 可能上升，也可能下降。

作为一个结论有：在回归中增加一个新变量，当且仅当新变量的

t 统计量在绝对值上大于 1 ，调整 R 方才会有所提高；

在回归中增加一组变量时，当且仅当这组新变量联合显著性的 F 统计量大于 1 ，调整 R 方才会有所提高。

Adjusted R-Squared (cont)

It’s easy to see that the adjusted R2 is just (1 – R2)(n – 1) / (n – k – 1), but most packages will give you both R2 and adj-R2

You can compare the fit of 2 models (with the same y) by comparing the adj-R2

You cannot use the adj-R2 to compare models with different y’s (e.g. y vs. ln(y))

Using adjusted R-squared to choose between nonnested models.

2 0.6211R

2 0.6226R

2 20.061, 0.03R R

2 20.148, 0.09R R

391732982salarySST

66.72lsalarySST

Controlling too many factors in regression analysis Important not to fixate too much on adj-R2

and lose sight of theory and common sense If economic theory clearly predicts a variable

belongs, generally leave it in Don’t want to include a variable that prohibits

a sensible interpretation of the variable of interest – remember ceteris paribus interpretation of multiple regression

在研究啤酒税对交通死亡率影响的回归模型中，是否应该将人均啤酒消费量变量包括在模型之中？

在保持 beercons 不变的情况下，死亡率因 tax 提高 1 个百分点而导致的差异。这一说法是否有意义？

Adding regressors to reduce the error of variance 在回归中增加一个新的自变量会加剧多重共线

性问题；另一方面，从误差项中取出一些因素作为解释变量可以减少误差方差。

应该将那些影响 y 而又与所有我们关心的自变量都无关的自变量包括进来。

Prediction and Residual Analysis Suppose we want to use our estimates to

obtain a specific prediction. First, suppose that we want an estimate of

E(y|x1=c1,…xk=ck) = 0 = 0+1c1+ …+ kck

This is easy to obtain by substituting the x’s in our estimated model with c’s , but what about a standard error?

Really just a test of a linear combination

Predictions (cont)

Can rewrite as 0 = 0 – 1c1 – … – kck

Substitute in to obtain y = 0 + 1 (x1 - c1) + … + k (xk - ck) + u

So, if you regress yi on (xij - cij) the intercept will give the predicted value and its standard error

Note that the standard error will be smallest when the c’s equal the means of the x’s

Example: CI for predicted college GPA

2.7 1.96 0.020 : 2.66 ~ 2.74CI

0 1200, 0 30, 0 5sat sat hsperc hsperc hsize hsize

Predictions (cont)

This standard error for the expected value is not the same as a standard error for an new outcome on y

We need to also take into account the variance in the unobserved error.

0 0 0 0 0 00 1 1 0

0 00 0 1 1

0 00 1 10 0

12 20 0 2

ˆ ˆ ˆ

ˆ ˆ ˆˆˆ ˆ ˆˆ

ˆ ˆ ,

ˆ ˆ ˆso

e y y x x u y

E y E E x E x

E e E u

Var e Var y Var u

e e se y

的方差有两个来源：第一个是的抽样误差，来自我们对

0ˆvar 1j jse c n y n

的估计。因为，所以与成比例。第二个是总体误差的方差，它不随样本容量的变化而变化。

ˆ ˆ ˆ 1

ˆ ˆ 0.95, 95%

0 00.025 0.025

e e se e n k t

P -t e se e t y

在经典线性模型（）假定下，和都是正态分布，所以也是正态分布的。服从一个自由度为的分布。

则的一个的预测区间：

0 00.025ˆ ˆy t se e

Prediction interval

for interval prediction 95% a have we

ˆˆ given that so ,~ˆˆ

yyetesee kn

Usually the estimate of s2 is much larger than the variance of the prediction, thus

This prediction interval will be a lot wider than the simple confidence interval for the prediction

Residual Analysis Information can be obtained from looking at

the residuals (i.e. predicted vs. observed)

Example: Regress price of cars on characteristics – big negative residuals indicate a good deal

Example: Regress average earnings for students from a school on student characteristics – big positive residuals indicate greatest value-added

ˆ ˆi i iu y y

例如， HPRICE1.RAW 的住房价格模型中。

0 1 2 3

81 81 81ˆ ˆ 120.206

price lotsize sqrft bdrms u

u price price

Predicting y in a log model

Simple exponentiation of the predicted ln(y) will underestimate the expected value of y

Instead need to scale this up by an estimate of the expected value of exp(u)

logexp

exp exp

| exp exp

In this case can predict y as followsˆˆ ˆexp 2 exp ln

exp( ) exp( 2) i

f ~ 0,

y x x uy x x u

E y x E u x x

E y x x x

ˆ ˆexp logy y

0 1 1 2 2

ˆ1ˆ ˆ(2)

ˆ ˆ(3)

ˆ4 :ˆ ˆ ˆ ˆˆlog

ˆ ˆˆ(5) exp log

1 2 k i

log y ylogy x ,x , ,x logy .

i, m = exp logy .y m m

x ,x , ,x logy

y x x xy y

当因变量为时对的预测：（）从对的回归中得到拟合值对每个观测都求出在不设截距的情况下求对的回归，的系数

（惟一的系数）就是的估计值。（）对于给定的的值，求出

利用直接得 ˆ.y到预测值

例 6.7 对 CEO 薪水的预测

ˆˆ exp

ˆˆ 1.117ˆ 4.504 0.163 log 5000 0.109 log 10000 0.0117 10 7.013ˆ 1.117 exp 7.013 1240.967 1240967

m lsalary

salary m

lsalary

salary

对样本中的每一个观测都求出；

将对进行回归（没有常数项）得。

或美元。

Comparing log and level models A by-product of the previous procedure is a

method to compare a model in logs with one in levels.

Take the fitted values from the auxiliary regression, and find the sample correlation between this and y

Compare the R2 from the levels regression with this correlation squared

ˆ2 2ˆ

例 6.8 对 CEO 薪水的预测

2ˆ0 ,

20 1 2 3

ˆ ˆˆ exp log , 0.493, 0.243

0.201salary salarysalary salary

salary salary

salary sales mktval ceoten u R

1 Copyright © 2007 Thomson Asia Pte. Ltd. All rights reserved. Multiple Regression Analysis:...

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