Multiple Regression by TBd92btU

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```									                        Multiple Regression                              7.1

• More than one explanatory/independent variable

yt  1   2 x2t   3 x3t  ...  k xkt  et
yt  E ( yt )  et
• This makes a slight change to the interpretation of the coefficients
• This changes the measure of degrees of freedom
• We need to modify one of the assumptions

EXAMPLE: trt = 1 + 2 pt + 3 at + e

EXAMPLE: qdt = 1 + 2 pt + 3 inct + et

EXAMPLE: gpat = 1 + 2 SATt + 3 STUDYt + et
Interpretation of Coefficient             7.2

yt  1   2 x2t   3 x3t  et
dy
 2
dx2   x3

dy
 3
dx3   x2

• 2 measures the change in Y from a change in X2, holding X3
constant.
• 23 measures the change in Y from a change in X3, holding X2
constant.
Assumptions of the Multiple Regression Model                    7.3

1. The Regression Model is linear in the parameters and error term
yt = 1 + 2 x2t + 3 x3t + … k xkt +et

2. Error Term has a mean of zero:
E(e) = 0  E(y) = 1 + 2 x2t + 3 x3t + … k xkt
3. Error term has constant variance: Var(e) = E(e2) = 2
4. Error term is not correlated with itself (no serial correlation):
Cov(ei,ej) = E(eiej) = 0 ij
5. Data on x’s are not random (and thus are uncorrelated with the
error term: Cov(X,e) = E(Xe) = 0) and they are NOT exact linear
functions of other explanatory variables.
6. (Optional) Error term has a normal distribution. E~N(0, 2)
Estimation of the Multiple Regression Model                     7.4

• Let’s use a model with 2 independent variables:
yt  1   2 x2t   3 x3t  et
• A scatterplot of points is now a scatter “cloud”. We want to fit the
best “line” through these points. In 3 dimensions, the line becomes
a plane.
• The estimated “line” and a residual are defined as before:

yt  b1  b2 x2t  b3 x3t
ˆ
et  yt  yt
ˆ         ˆ
• The idea is to choose values for b1, b2, and b3 such that the sum of
squared residuals is minimized.
et2   ( yt  yt ) 2
ˆ              ˆ                                                            7.5

  ( yt  b1  b2 x2t  b3 x3t ) 2

From here, we minimize this expression with respect to b1,
b2, and b3. We set these three derivatives equal to zero and
Solve for b1, b2, b3. We get the following formulas:

b2 
 yt* x2t  x3t2   yt* x3t  x2t x3t
*     *            *     * *

   *
x22
t     x3t2
*
* * 2
x2t x3t

b3 
 yt* x3t  x22   yt* x2t  x2t x3t
*     *
t
*     * *                       Where:

 x2t  x3t   x2t x3t                                   ( yt  y )
* * 2                            *
*2    *2                                       yt
x2t  ( x2t  x2 )
*

b1  y  b2 x2  b3 x3
x3t  ( x3t  x3 )
*
[What is going on here? In the formula for b2, notice that if      7.6
x3 where omitted from the model, the formula reduces to the
familiar formula from Chapter 3.]

You may wonder why the multiple regression formulas on slide 7.5
aren’t equal to:

b2 
 ( yt  y )(x 2t  x2 )
    ( x 2t  x2 ) 2

b3 
 ( yt  y )(x3t  x3 )
    ( x 3t  x3 ) 2
We can use a Venn diagram to illustrate the idea of Regression as   7.7
Analysis of Variance

For Bivariate (Simple) Regression

y
x

For Multiple Regression

y
x2

x3
Example of Multiple Regression                           7.8

Suppose we want to estimate a model of home prices using data
on the size of the house (sqft), the number of bedrooms (bed) and the
number of bathrooms (bath). We get the following results:

ˆ
pri cet  129.062  0.1548sqftt  21.588bedt 12.193batht

How does a negative coefficient estimate on bed and bath make
sense?
7.9
Expected Value

E (b1 )  1                      We will omit the proofs. The Least
E (b2 )   2                     Squares estimator for multiple regression
is unbiased, regardless of the number of
E (b3 )   3                     independent variables

Variance Formulas With 2 Independent Variables

2                   Where r23 is the correlation
Var(b2 ) 
(1  r23 ) ( x2t  x2 ) 2
2                         between x2 and x3 and the
parameter 2 is the variance
2                   of the error term.
Var(b3 ) 
(1  r23 ) ( x3t  x3 ) 2
2

2
We need to estimate using the formula                  
2    ˆ
et2
This estimate has T-k degrees of freedom.                    T k
Gauss Markov Theorem                                 7.10

Under the assumptions 1-5 (the 6th assumption isn’t needed for the
theorem to be true) of the linear regression model, the least squares
estimators b1, b2, …bk have the smallest variance of all linear and
unbiased estimators of 1 , 2,… k. They are the BLUE (Best,
linear, unbiased, estimator)
Confidence Intervals and Hypothesis Testing                          7.11

• The methods for constructing confidence intervals and conducting
hypothesis tests are the same as they were for simple regression.

• The format for a confidence interval is:

bi  tc se(bi )
Where tc depends on the level of confidence and has T-k degrees of
freedom. T is the number of observations and k is the number of
independent variables plus one for the intercept.

• Hypothesis Tests:
t
bi   i 
Ho : i = c
H1 : i  c
se(bi )
Use the value of c for i when calculating t. If t > tc or t < - tc  reject Ho
If c is 0, then we call it a test of significance.
Goodness of Fit                                  7.12

• R2 measures the proportion of the variance in the dependent
variable that is explained by the independent variable. Recall that

R 
2 SSR
 1
SSE
 1
 et2ˆ
SST      SST       ( yt  y ) 2

• Least Squares chooses the line that produces the smallest sum of
squared residuals, it also produces the line with the largest R2. It
also has the property that the inclusion of additional independent
variables will never increase and will often lower the sum of
squared residuals, meaning that R2 will never fall and will often
increase when new independent variables are added, even if the
variables have no economic justification.

R  1
2            et2 /(T  k )
ˆ
 ( yt  y ) 2 /(T  1)

A sample of 55 JMU students was taken Fall 2002. Data on
• GPA
• SAT scores
• Credit Hours Completed
• Hours of Study per Week
• Hours at a Job per week
• Hours at Extracurricular Activites
Three models were estimated:
gpat = 1 + 2 SATt + et

gpat = 1 + 2 SATt + 3 CREDITSt + 4 STUDYt + 5JOBt + 6 ECt + et

gpat = 1 + 2 SATt + 3 CREDITSt + 4 STUDYt + 5JOBt + et
7.14
Regression Statistics
Here is our simple   Multiple R             0.270081672
Regression model.    R Square               0.072944109
Standard Error         0.455494651
Observations                    55

Coefficients Standard Error   t Stat     P-value
Intercept            1.799734629    0.631013633 2.852132721 0.006178434
SAT Total            0.001057599    0.000517894 2.042114498 0.046129915

Regression Statistics
Here is our multiple     Multiple R             0.465045986
regression model.        R Square               0.216267769
Both R2 and              Standard Error            0.4355661
increased with the
inclusion of                                   Coefficients Standard Error      t Stat       P-value
4 additional indep.      Intercept             1.538048132    0.686920438    2.239048435   0.029726041
SAT Total             0.000943615    0.000503684    1.873427614   0.066980023
variables.                                     0.003382201
Credit Hours (completed)             0.002664489    1.269361823   0.210308107
Study (Hrs/wk)        0.010762866    0.006782587    1.586837918   0.118982276
Job (Hrs/wk)           -0.00795843   0.006452798    -1.23333021    0.22333641
EC (Hrs/wk)           0.002606617    0.009216993    0.282805582   0.778517157
7.15
Regression Statistics
Multiple R             0.463668569
Notice that the      R Square               0.214988542
Exclusion of EC      Standard Error         0.431540195

R2 but reduces
R2                   Intercept
Coefficients Standard Error
1.53616637   0.680539354
t Stat
2.257277792
P-value
0.028390614
SAT Total             0.000951057    0.000498346    1.908425924   0.062085531
0.003441283
Credit Hours (completed)             0.002631734    1.307610208   0.196986336
Study (Hrs/wk)        0.011267319    0.006483348    1.737885925   0.088386942
Job (Hrs/wk)             -0.0080306  0.006388155    -1.25710723   0.214555464

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