Ka-fu Wong © 2003 Intro 1- 1 Dr. Ka-fu Wong ECON1003 Analysis of Economic Data.
Ka-fu Wong © 2003 Chap 11- 1 Dr. Ka-fu Wong ECON1003 Analysis of Economic Data.
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Transcript of Ka-fu Wong © 2003 Chap 11- 1 Dr. Ka-fu Wong ECON1003 Analysis of Economic Data.
Ka-fu Wong © 2003 Chap 11- 1
Dr. Ka-fu Wong
ECON1003Analysis of Economic Data
Ka-fu Wong © 2003 Chap 11- 2l
GOALS
1. Draw a scatter diagram.2. Understand and interpret the terms dependent
variable and independent variable.3. Calculate and interpret the coefficient of correlation,
the coefficient of determination, and the standard error of estimate.
4. Conduct a test of hypothesis to determine if the population coefficient of correlation is different from zero.
5. Calculate the least squares regression line and interpret the slope and intercept values.
6. Construct and interpret a confidence interval and prediction interval for the dependent variable.
7. Set up and interpret an ANOVA table.
Chapter ElevenLinear Regression and CorrelationLinear Regression and Correlation
Ka-fu Wong © 2003 Chap 11- 3
Correlation Analysis
Correlation Analysis is a group of statistical techniques used to measure the strength of the association between two variables.
A Scatter Diagram is a chart that portrays the relationship between the two variables.
The Dependent Variable is the variable being predicted or estimated.
The Independent Variable provides the basis for estimation. It is the predictor variable.
Ka-fu Wong © 2003 Chap 11- 4
Types of Relationships
Direct vs. InverseDirect - X and Y increase together Inverse - X and Y have opposite
directions Linear vs. Curvilinear
Linear - Straight line best describes the relationship between X and Y
Curvilinear - Curved line best describes the relationship between X and Y
Ka-fu Wong © 2003 Chap 11- 5
Direct vs. Inverse Relationship
Advertising
Sa
les
Anti-Pollution ExpendituresP
ollu
tio
n E
mis
sio
ns
Positive Slope
Negative Slope
Direct Relationship Inverse Relationship
Ka-fu Wong © 2003 Chap 11- 6
Example
Suppose a university administrator wishes to determine whether any relationship exists between a student’s score on an entrance examination and that student’s cumulative GPA. A sample of eight students is taken. The results are shown below
Student Exam Score GPA
A 74 2.6
B 69 2.2
C 85 3.4
D 63 2.3
E 82 3.1
F 60 2.1
G 79 3.2
H 91 3.8
Ka-fu Wong © 2003 Chap 11- 7
Scatter Diagram: GPA vs. Exam Score
| | | | | | | | | |
50 55 60 65 70 75 80 85 90 95
Exam Score
4.00 -3.75 -3.50 -3.25 -3.00 -2.75 -2.50 -2.25 -2.00 -
Cu
mu
lati
ve G
PA
Ka-fu Wong © 2003 Chap 11- 8
Possible relationships between X and Y in Scatter Diagrams
Y
X
(a) Direct linearY
X
(b) Inverse linear
Y
X
(f) No relationship
Y
X
(c) Direct curvilinear
Y
X
(d) Inverse curvilinearY
X
(e) Inverse linearwith more scattering
Ka-fu Wong © 2003 Chap 11- 9
The Coefficient of Correlation, r
The Coefficient of Correlation (r) is a measure of the strength of the linear relationship between two variables. It requires interval or ratio-scaled data. It can range from -1.00 to 1.00. Values of -1.00 or 1.00 indicate perfect and
strong correlation. Values close to 0.0 indicate weak
correlation. Negative values indicate an inverse
relationship and positive values indicate a direct relationship.
Ka-fu Wong © 2003 Chap 11- 10
Formula for r
n
1i
2i
n
1i
2i
n
1iii
n
1i
2i
n
1i
2i
n
1iii
yx
2xy
)yy()xx(
)yy)(xx(
)1n(
)yy(
)1n(
)xx(
)1n(
)yy)(xx(
ss
sr
We calculate the coefficient of correlation from the following formulas.
Sample covariance between x and y
Sample standard deviation of x
Sample standard deviation of y
Ka-fu Wong © 2003 Chap 11- 11
Perfect Negative Correlation (r = -1)
0 1 2 3 4 5 6 7 8 9 10
10 9 8 7 6 5 4 3 2 1 0
X
Y
Ka-fu Wong © 2003 Chap 11- 12
Perfect Positive Correlation (r = +1)
X
Y
10 9 8 7 6 5 4 3 2 1 0
0 1 2 3 4 5 6 7 8 9 10
Ka-fu Wong © 2003 Chap 11- 13
Zero Correlation (r = 0)
X
Y
0 1 2 3 4 5 6 7 8 9 10
10 9 8 7 6 5 4 3 2 1 0
Ka-fu Wong © 2003 Chap 11- 14
Strong Positive Correlation (0<r<1)
X
Y
10 9 8 7 6 5 4 3 2 1 0
0 1 2 3 4 5 6 7 8 9 10
Ka-fu Wong © 2003 Chap 11- 15
Coefficient of Determination
The coefficient of determination (r2) is the proportion of the total variation in the dependent variable (Y) that is explained or accounted for by the variation in the independent variable (X). It is the square of the coefficient of correlation. It ranges from 0 to 1. It does not give any information on the direction
of the relationship between the variables.
Special cases: No correlation: r=0, r2=0. Perfect negative correlation: r=-1, r2=1. Perfect positive correlation: r=+1, r2=1.
Ka-fu Wong © 2003 Chap 11- 16
EXAMPLE 1
Dan Ireland, the student body president at Toledo State University, is concerned about the cost to students of textbooks. He believes there is a relationship between the number of pages in the text and the selling price of the book. To provide insight into the problem he selects a sample of eight textbooks currently on sale in the bookstore. Draw a scatter diagram. Compute the correlation coefficient.Book Page Price ($)
Intro to History 500 84
Basic Algebra 700 75
Intro to Psyc 800 99
Intro to Sociology 600 72
Bus. Mgt. 400 69
Intro to Biology 500 81
Fund. of Jazz 600 63
Princ. Of Nursing 800 93
Ka-fu Wong © 2003 Chap 11- 17
Example 1 continued
400 500 600 700 800
60
70
80
90
100
Page
Scatter Diagram of Number of Pages and Selling Price of Text
Price ($)
Ka-fu Wong © 2003 Chap 11- 18
Example 1 continued
Book Page Price ($)
X Y
Intro to History 500 84
Basic Algebra 700 75
Intro to Psyc 800 99
Intro to Sociology
600 72
Bus. Mgt. 400 69
Intro to Biology 500 81
Fund. of Jazz 600 63
Princ. Of Nursing
800 93
Total 4,900
636
The correlation between the number of pages and the selling price of the book is 0.614. This indicates a moderate association between the variable.
n
1i
2i
n
1i
2i
n
1iii
)yy()xx(
)yy)(xx(r
Ka-fu Wong © 2003 Chap 11- 19
EXAMPLE 1 continued
Is there a linear relation between number of pages and price of books?
Test the hypothesis that there is no correlation in the population. Use a .02 significance level.
Under the null hypothesis that there is no correlation in the population. The statistic
)2/()1( 2
nr
rt
follows student t-distribution with (n-2) degree of freedom.
Ka-fu Wong © 2003 Chap 11- 20
EXAMPLE 1 continued
Step 1: H0: The correlation in the population is zero. H1: The correlation in the population is not zero.
Step 2: H0 is rejected if t>3.143 or if t<-3.143. There are 6 degrees of freedom, found by n – 2 = 8 – 2 = 6.
Step 3: To find the value of the test statistic we use:
905.1)614(.1
28614.
)2/()1( 22
nr
rt
Step 4: H0 is not rejected. We cannot reject the hypothesis that there is no correlation in the population. The amount of association could be due to chance.
Ka-fu Wong © 2003 Chap 11- 21
Regression Analysis
In regression analysis we use the independent variable (X) to estimate the dependent variable (Y).
The relationship between the variables is linear.
Both variables must be at least interval scale.
Ka-fu Wong © 2003 Chap 11- 22
Simple Linear Regression Model
Relationship Between Variables Is a Linear Function
iii XY 10
Y intercept Slope Random Error
Dependent (Response) Variable
Independent (Explanatory) Variable x
y
0 Run
Rise
1 = Rise/Run
0 and 1 are unknown,therefore, are estimated from the data.
Ka-fu Wong © 2003 Chap 11- 23
Finance Application: Market Model
One of the most important applications of linear regression is the market model.
It is assumed that rate of return on a stock (R) is linearly related to the rate of return on the overall market (Rm).
Rate of return on a particular stock
Rate of return on some major stock index
The beta coefficient measures how sensitive the stock’s rate of return is to changes in the level of the overall market.
R = 0 + 1Rm +
Ka-fu Wong © 2003 Chap 11- 24
Assumptions Underlying Linear Regression
For each value of X, there is a group of Y values, and these Y values are normally distributed.
The means of these normal distributions of Y values all lie on the straight line of regression.
The standard deviations of these normal distributions are equal.
The Y values are statistically independent. This means that in the selection of a sample, the Y values chosen for a particular X value do not depend on the Y values for any other X values.
Ka-fu Wong © 2003 Chap 11- 25
Choosing the line that fits best
The estimates are determined by drawing a sample from the population of interest, calculating sample statistics. producing a straight line that cuts into the data.
The question is:Which straight line fits best?
x
y
Ka-fu Wong © 2003 Chap 11- 263
3
41
1
4
(1,2)
2
2
(2,4)
(3,1.5)
Sum of squared differences = (2 - 1)2 + (4 - 2)2 + (1.5 - 3)2 +
(4,3.2)
(3.2 - 4)2 = 6.89Sum of squared differences = (2 -2.5)2 + (4 - 2.5)2 + (1.5 - 2.5)2 + (3.2 - 2.5)2 = 3.99
2.5
Let us compare two lines
The second line is horizontal
The smaller the sum of squared differences the better the fit of the line to the data. That is, the line with the least sum of squares (of differences) will fit the line best.
The best line is the one that minimizes the sum of squared vertical differences between the points and the line.
Choosing the line that fits best
Ka-fu Wong © 2003 Chap 11- 27
Choosing the line that fits bestOrdinary Least Squares (OLS) Principle
Straight lines can be described generally by Y = b0 + b1X
Finding the best line with smallest sum of squared difference is the same as
∑≡n
1i
2i10i10
b,b
)]xb(b[y )b,S(bmin10
Let b0* and b1
* be the solution of the above
problem. Y* = b0* + b1
*Xis known as the “average predicted value” (or simply “predicted value”) of y for any X.
Ka-fu Wong © 2003 Chap 11- 28
Coefficient estimates from the ordinary least squares (OLS) principle
Solving the minimization problem implies the first order conditions:
0))](-xxb(b-2[y b
)b,S(b
0)](-1)xb(b-2[y b
)b,S(b
)]xb(b -[y )b,S(b
n
1iii10i
1
10
n
1ii10i
0
10
n
1i
2i10i10
∑∂
∂
∑∂
∂
∑≡
Ka-fu Wong © 2003 Chap 11- 29
Coefficient estimates from the ordinary least squares (OLS) principle
Solving the first order conditions implies
xbyn
xb
n
yb
SS
SS
)x(x
)y)(yx-(x
xnx
yxnyx
)x()xn(
yx)yxn(b
1
n
1ii
1
n
1ii
0
xx
xy
2n
1ii
n
1iii
2n
1i
2i
n
1iii
2n
1ii
n
1i
2i
n
1ii
n
1ii
n
1iii
1
Ka-fu Wong © 2003 Chap 11- 30
EXAMPLE 2 continued from Example 1
Develop a regression equation for the information given in EXAMPLE 1. The information there can be used to estimate the selling price based on the number of pages.
05143.)900,4()000,150,3(8
)636)(900,4()200,397(821
b
0.488
900,405143.0
8
6360 b
Ka-fu Wong © 2003 Chap 11- 31
Example 2 continued from Example 1
The regression equation is:
Y* = 48.0 + .05143X
The equation crosses the Y-axis at $48. A book with no pages would cost $48.
The slope of the line is .05143. Each additional page costs about $0.05 or five cents.
The sign of the b value and the sign of r will always be the same.
Ka-fu Wong © 2003 Chap 11- 32
Example 2 continued from Example 1
We can use the regression equation to estimate values of Y.
The estimated selling price of an 800 page book is $89.14, found by
Y* = 48.0 + .05143X = 48.0 + .05143(800) = 89.14
Ka-fu Wong © 2003 Chap 11- 33
Standard Error of Estimate (denoted se or Sy.x)
Measures the reliability of the estimating equation A measure of dispersion Measures the variability, or scatter of the observed
values around the regression line
2
)(
2
)(
11
10
2
1
2
1
*
.
n
yxbyby
n
yyss
n
iii
n
ii
n
ii
n
iii
xye
Ka-fu Wong © 2003 Chap 11- 34
More Accurate Estimatorof X, Y Relationship
Less Accurate Estimatorof X, Y Relationship
Scatter Around the Regression Line
Ka-fu Wong © 2003 Chap 11- 35
se measures the dispersion of the points around the regression line If se = 0, equation is a “perfect” estimator
se is used to compute confidence intervals of the estimated value
Assumptions: Observed Y values are normally
distributed around each estimated value of Y*
Constant variance
Interpreting the Standard Error of the Estimate
Ka-fu Wong © 2003 Chap 11- 36
X1
X2
X
Y
f(e)y values are normally distributed around the regression line.
For each x value, the “spread” or variance around the regression line is the same.
Regression Line
Variation of Errors Around the Regression Line
Ka-fu Wong © 2003 Chap 11- 37
De
pen
de
nt
Va
ria
ble
( Y
)
Independent Variable (X)
Y = b0 + b1X + 2se
Y = b0 + b1X + 1se
Y = b0 + b1X - 1se
Y = b0 + b1X - 2se
Y = b0 + b1X regression line
2se (95.5% Lie in this Region)
Scatter around the Regression Line
1se (68% Lie in this Region)
Ka-fu Wong © 2003 Chap 11- 38
Example 3 continued from Example 1 and 2.
Find the standard error of estimate for the problem involving the number of pages in a book and the selling price.
408.1028
)200,397(05143.0)636(48606,51
2
)(1
11
02
1
n
yxbybys
n
iii
n
ii
n
ii
e
Ka-fu Wong © 2003 Chap 11- 39
Equations for the Interval Estimates
htsy e*
2
1
2
)(
)(1
xx
xx
nh
n
ii
htsy e 1*
Confidence Interval for the Mean of y
Prediction Interval for the Mean of y
Ka-fu Wong © 2003 Chap 11- 40
Confidence Interval Estimate for Mean Response
X
The following factors influence the width of the interval: Std Error, Sample Size, X Value
y* = b0+b1xi
Ka-fu Wong © 2003 Chap 11- 41
Confidence Interval continued from Example 1, 2 and 3.
For books of 800 pages long, what is that 95% confidence interval for the mean price? This calls for a confidence interval on the
average price of books of 800 pages long.
31.1514.898
)4900(000,150,3
)5.612800(
8
1)408.10(447.214.89
)(
)(1
2
2
2
1
2**
xx
xx
ntsyhtsy
n
ii
ee
Ka-fu Wong © 2003 Chap 11- 42
Prediction Interval continued from Example 1, 2 and 3.
For a book of 800 pages long, what is the 95% prediction interval for its price? This calls for a prediction interval on the price
of an individual book of 800 pages long.
72.2914.898
)4900(000,150,3
)5.612800(
8
11)408.10(447.214.89
)(
)(111
2
2
2
1
2**
xx
xx
ntsyhtsy
n
ii
ee
Ka-fu Wong © 2003 Chap 11- 43
1. Slope (b1)
Estimated Y changes by b1 for each 1 unit increase in X
2. Y-Intercept (b0 ) Estimated value of Y when X = 0
Interpretation of Coefficients
Ka-fu Wong © 2003 Chap 11- 44
1. Tests if there is a linear relationship between X & Y
2. Involves population slope
3. Hypotheses H0: = 0 (no linear relationship)
H1: 0 (linear relationship)
4. Theoretical basis is sampling distribution of slopes
Test of Slope Coefficient (b1)
Ka-fu Wong © 2003 Chap 11- 45
Sampling Distribution of the Least Squares Coefficient Estimator
If the standard least squares assumptions hold, then b1 is an unbiased estimator of 1 and has a population variance
2
2
1
2
22
)1()(
1
xn
ii
b snxx
and an unbiased sample variance estimator
2
2
1
2
22
)1()(
1
x
en
ii
eb sn
s
xx
ss
Bias = E(b1) 1 “ Unbiasd” means E(b1) 1 =0
Ka-fu Wong © 2003 Chap 11- 46
Basis for Inference About the Population Regression Slope
Let 1 be a population regression slope and b1 its least squares estimate based on n pairs of sample observations. Then, if the standard regression assumptions hold and it can also be assumed that the errors i are normally distributed, the random variable
1
11 -
bs
bt
is distributed as Student’s t with (n – 2) degrees of freedom. In addition the central limit theorem enables us to conclude that this result is approximately valid for a wide range of non-normal distributions and large sample sizes, n.
Ka-fu Wong © 2003 Chap 11- 47
Tests of the Population Regression Slope
If the regression errors i are normally distributed and the standard least squares assumptions hold (or if the distribution of b1 is approximately normal), the following tests have significance value :
1. To test either null hypothesis H0: 1 = 1
* or H0:1 1*
against the alternative H1: 1 > 1
*
The decision rule is to reject if
2),(n-b
*11 t
sβ-b
t1
Ka-fu Wong © 2003 Chap 11- 48
Tests of the Population Regression Slope
2. To test either null hypothesis H0: 1 = 1
* or H0:1 > 1*
against the alternative H1: 1 1
*
the decision rule is to reject if
2),(nb
*11 t
sβ-b
t1
-≤
Ka-fu Wong © 2003 Chap 11- 49
Tests of the Population Regression Slope
3. To test either null hypothesis H0: 1 = 1
* against the alternative H1: 1 1
*
the decision rule is to reject if
/22),(nb
*11
/22),(nb
*11 t-
sβ-b
t or t s
β-bt
11
-- ≤≥
/22),(n-b
*11 t
sβb
t 1
≥ -
Equivalently
Ka-fu Wong © 2003 Chap 11- 50
Confidence Intervals for the Population Regression Slope 1
If the regression errors i , are normally distributed and the standard regression assumptions hold, a 100(1 - )% confidence interval for the population regression slope 1 is given by
11 b/22),(n11b/22),(n-1 stbβst-b -
Ka-fu Wong © 2003 Chap 11- 51
Significance Test and Estimation for Slope
If the regression assumptions hold, we can reject H0: 1 = 0 at the level of significance (probability of Type I error equal to ) if and only if the appropriate rejection point condition holds or, equivalently, if the corresponding p-value is less than .
xx
bb SS
ss
s
bt=
1
1
where1
Test Statistic
0:
0:
0:
1
1
1
a
a
a
H
H
H
2/2/
2/
or
isthat,
tttt
tt
tt
tt
t, t/2 and p-values are based on n – 2 degrees of freedom.
Alternative Reject H0 if: p-Value
tofrightondistributit underarea Twice
tofleftondistributit underArea
tofrightondistributit underArea
100(1-)% Confidence Interval for 1
][12/1 bstb
Ka-fu Wong © 2003 Chap 11- 52
Significance Test and Estimation for y-Intercept
If the regression assumptions hold, we can reject H0: 0 = 0 at the level of significance (probability of Type I error equal to ) if and only if the appropriate rejection point condition holds or, equivalently, if the corresponding p-value is less than .
xxb
b SS
x
nss
s
bt=
20 1
where0
0
Test Statistic
0:
0:
0:
0
0
0
a
a
a
H
H
H
2/2/
2/
or
isthat,
tttt
tt
tt
tt
t, t/2 and p-values are based on n – 2 degrees of freedom.
Alternative Reject H0 if: p-Value
tofrightondistributit underarea Twice
tofleftondistributit underArea
tofrightondistributit underArea
100(1-)% Conf Interval for 0
][02/0 bstb
Ka-fu Wong © 2003 Chap 11- 53
Some cautions about the interpretation of significance tests
Rejecting H0: b1 = 0 and concluding that the relationship between x and y is significant does not enable us to conclude that a cause-and-effect relationship is present between x and y.
Causation requires: Association Accurate time sequence Other explanation for correlation
Correlation Causation Correlation Causation
Ka-fu Wong © 2003 Chap 11- 54
Some cautions about the interpretation of significance tests
Just because we are able to reject H0: 1 = 0 and demonstrate statistical significance does not enable us to conclude that there is a linear relationship between x and y.
Linear relationship is a very small subset of possible relationship among variables.
A test of linear versus nonlinear relationship requires another batch of analysis.
Ka-fu Wong © 2003 Chap 11- 55
Variation Measures Coeff. Of Determination Standard Error of Estimate
Test Coefficients for Significance
yi* = b0 +b1xi
Evaluating the Model
Ka-fu Wong © 2003 Chap 11- 56
Y
X
Y
Xi
Total Sum of Squares (Yi - Y)2
Unexplained Sum of Squares (Yi -Yi
*)2
Explained Sum of Squares (Yi
* - Y)2
Yi
SST
SSE
SSR
yi* = b0 +b1xi
Variation Measures
Ka-fu Wong © 2003 Chap 11- 57
Total Sum of Squares (SST)Measures variation of observed Yi
around the mean,Y Explained Variation (SSR)
Variation due to relationship between X & Y
Unexplained Variation (SSE)Variation due to other factors
SST=SSR+SSE
Measures of Variation in Regression
Ka-fu Wong © 2003 Chap 11- 58
R2 (=r2, the coefficient of determination)
measures the proportion of the variation in y that is explained by the variation in x.
n
ii
n
ii
n
ii
n
ii yy
SSR
yy
SSEyy
yy
SSER
1
2
1
2
1
2
1
2
2
)()(
)(
)(1
R2 takes on any value between zero and one. R2 = 1: Perfect match between the line and the data
points. R2 = 0: There are no linear relationship between x and
y.
Variation in y (SST) = SSR + SSE
Ka-fu Wong © 2003 Chap 11- 59
Summarizing the Example’s results (Example 1, 2 and 3)
The estimated selling price for a book with 800 pages is $89.14.
The standard error of estimate is $10.41. The 95 percent confidence interval for all
books with 800 pages is $89.14 ± $15.31. This means the limits are between $73.83 and $104.45.
The 95 percent prediction interval for a particular book with 800 pages is $89.14 ± $29.72. The means the limits are between $59.42 and $118.86.
These results appear in the following output.
Ka-fu Wong © 2003 Chap 11- 60
Example 3 continued
Regression Analysis: Price versus Pages
The regression equation isPrice = 48.0 + 0.0514 Pages
Predictor Coef SE Coef T PConstant 48.00 16.94 2.83 0.030Pages 0.05143 0.02700 1.90 0.105
S = 10.41 R-Sq = 37.7% R-Sq(adj) = 27.3%
Analysis of VarianceSource DF SS MS F PRegression 1 393.4 393.4 3.63 0.105Residual Error 6 650.6 108.4Total 7 1044.0
Ka-fu Wong © 2003 Chap 11- 61
Testing for Linearity
Key Argument: If the value of y does not change linearly
with the value of x, then using the mean value of y is the best predictor for the actual value of y. This implies is preferable.
If the value of y does change linearly with the value of x, then using the regression model gives a better prediction for the value of y than using the mean of y. This implies is preferable.
yy
* yy
Ka-fu Wong © 2003 Chap 11- 62
Three Tests for Linearity
Testing the Coefficient of Correlation
H0: = 0 There is no linear relationship between x and y.
H1: 0 There is a linear relationship between x and y.
Test Statistic:
n
iie xnxs
bt
1
22
1
/
Testing the Slope of the Regression LineH0: = 0 There is no linear relationship between x and y.
H1: 0 There is a linear relationship between x and y.
Test Statistic:
)2/()1( 2
nr
rt
Ka-fu Wong © 2003 Chap 11- 63
Three Tests for Linearity
The Global F-testH0: There is no linear relationship between x and y.
H1: There is a linear relationship between x and y.
Test Statistic:
)2/()(
)(
)2/(
1/
2
1
*
2
1
*
nyy
yy
nSSE
SSR
MSE
MSRF
n
iii
n
ii
[Variation in y] = SSR + SSE. Large F results from a large SSR. Then, much of the variation in y is explained by the regression model. The null hypothesis should be rejected; thus, the model is valid.Note: At the level of simple linear regression, the global F-test is equivalent to the t-test on b1. When we conduct regression analysis of multiple variables, the global F-test will take on a unique function.
Ka-fu Wong © 2003 Chap 11- 64
PurposesExamine Linearity Evaluate violations of assumptions
Graphical Analysis of ResidualsPlot residuals versus Xi values
Difference between actual Yi & predicted Yi
*
Studentized residuals:Allows consideration for the
magnitude of the residuals
Residual Analysis
Ka-fu Wong © 2003 Chap 11- 65
Residual Analysis for Linearity
Not Linear LinearOK
X
e e
X
Ka-fu Wong © 2003 Chap 11- 66
Heteroscedasticity OK Homoscedasticity
Using Standardized Residuals
SR
X
SR
X
Residual Analysis for Homoscedasticity
When the requirement of a constant variance (homoscedasticity) is violated we have heteroscedasticity.
Ka-fu Wong © 2003 Chap 11- 67
Residual Analysis for Independence
Not Independent Independent
X
SR
X
SR
OK
Ka-fu Wong © 2003 Chap 11- 68
A time series is constituted if data were collected over time.
Examining the residuals over time, no pattern should be observed if the errors are independent.
When a pattern is detected, the errors are said to be autocorrelated.
Autocorrelation can be detected by graphing the residuals against time.
Non-independence of error variables
Ka-fu Wong © 2003 Chap 11- 69
+
+++ +
++
++
+ +
++ + +
+
++ +
+
+
+
+
+
+Time
Residual Residual
Time+
+
+
Note the runs of positive residuals,replaced by runs of negative residuals
Note the oscillating behavior of the residuals around zero.
0 0
Patterns in the appearance of the residuals over time indicates that autocorrelation exists.
Ka-fu Wong © 2003 Chap 11- 70
n
ii
n
iii
e
eeD
1
2
2
21)( Should be close to 2.
If not, examine the model for autocorrelation.
Used when data is collected over time to detect autocorrelation (Residuals in one time period are related to residuals in another period)
Measures Violation of independence assumption
The Durbin-Watson Statistic
Ka-fu Wong © 2003 Chap 11- 71
An outlier is an observation that is unusually small or large.
Several possibilities need to be investigated when an outlier is observed:There was an error in recording the value.The point does not belong in the sample.The observation is valid.
Identify outliers from the scatter diagram. It is customary to suspect an observation is
an outlier if its |standard residual| > 2
Outliers
Ka-fu Wong © 2003 Chap 11- 72
+
+
+
+
+ +
+ + ++
+
+
+
+
+
+
+
The outlier causes a shift in the regression line
… but, some outliers may be very influential
++++++++++
An outlier An influential observation
Ka-fu Wong © 2003 Chap 11- 73
Nonnormality or heteroscedasticity can be remedied using transformations on the y variable.
The transformations can improve the linear relationship between the dependent variable and the independent variables.
Many computer software systems allow us to make the transformations easily.
Remedying violations of the required conditions
Ka-fu Wong © 2003 Chap 11- 74
A brief list of transformations
Y’ = log y (for y > 0) Use when the se increases with y, or
Use when the error distribution is positively skewed
y’ = y2
Use when the se2 is proportional to E(y), or
Use when the error distribution is negatively skewed
y’ = y1/2 (for y > 0) Use when the se
2 is proportional to E(y)
y’ = 1/y Use when se
2 increases significantly when y increases beyond some value.
Ka-fu Wong © 2003 Chap 11- 75
Example: Transformation to get linearity
Not Linear OK
X
e
Linear
e
X
Yi = b0 + b Xi + ei Yi = b0 + b1 Xi + b2 Xi2 +
ei
Ka-fu Wong © 2003 Chap 11- 76
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Chapter ElevenLinear Regression and CorrelationLinear Regression and Correlation