Numerical Analysis 2000 : Linear Algebra - Linear Systems and Eigenvalues (Numerical Analysis 2000)
Residual Analysis Purposes –Examine Functional Form (Linear vs. Non- Linear Model) –Evaluate...
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![Page 1: Residual Analysis Purposes –Examine Functional Form (Linear vs. Non- Linear Model) –Evaluate Violations of Assumptions Graphical Analysis of Residuals.](https://reader036.fdocuments.us/reader036/viewer/2022062503/5a4d1ae17f8b9ab05997717e/html5/thumbnails/1.jpg)
Residual AnalysisResidual Analysis
![Page 2: Residual Analysis Purposes –Examine Functional Form (Linear vs. Non- Linear Model) –Evaluate Violations of Assumptions Graphical Analysis of Residuals.](https://reader036.fdocuments.us/reader036/viewer/2022062503/5a4d1ae17f8b9ab05997717e/html5/thumbnails/2.jpg)
• Purposes– Examine Functional Form (Linear vs. Non-
Linear Model)– Evaluate Violations of Assumptions
• Graphical Analysis of Residuals
Residual AnalysisResidual Analysis
![Page 3: Residual Analysis Purposes –Examine Functional Form (Linear vs. Non- Linear Model) –Evaluate Violations of Assumptions Graphical Analysis of Residuals.](https://reader036.fdocuments.us/reader036/viewer/2022062503/5a4d1ae17f8b9ab05997717e/html5/thumbnails/3.jpg)
(X1, Y1)
For one value X1, a population contains may Y values. Their mean is Y1.
X1
Y
![Page 4: Residual Analysis Purposes –Examine Functional Form (Linear vs. Non- Linear Model) –Evaluate Violations of Assumptions Graphical Analysis of Residuals.](https://reader036.fdocuments.us/reader036/viewer/2022062503/5a4d1ae17f8b9ab05997717e/html5/thumbnails/4.jpg)
Y
X
A Population Regression Line
Y = X
![Page 5: Residual Analysis Purposes –Examine Functional Form (Linear vs. Non- Linear Model) –Evaluate Violations of Assumptions Graphical Analysis of Residuals.](https://reader036.fdocuments.us/reader036/viewer/2022062503/5a4d1ae17f8b9ab05997717e/html5/thumbnails/5.jpg)
Y
x
A Sample Regression Line
The sample line approximates the population regression line.
y = a + bx
![Page 6: Residual Analysis Purposes –Examine Functional Form (Linear vs. Non- Linear Model) –Evaluate Violations of Assumptions Graphical Analysis of Residuals.](https://reader036.fdocuments.us/reader036/viewer/2022062503/5a4d1ae17f8b9ab05997717e/html5/thumbnails/6.jpg)
Histogram of Y Values at X = X1
Y
f(e)
XX1
Y = XY1 = X1
![Page 7: Residual Analysis Purposes –Examine Functional Form (Linear vs. Non- Linear Model) –Evaluate Violations of Assumptions Graphical Analysis of Residuals.](https://reader036.fdocuments.us/reader036/viewer/2022062503/5a4d1ae17f8b9ab05997717e/html5/thumbnails/7.jpg)
Normal Distribution of Y Values when X = X1
Y
f(e)
XX1
Y1 = X1 Y = X
The standard deviation of the normal distribution is the standard error of estimate.
![Page 8: Residual Analysis Purposes –Examine Functional Form (Linear vs. Non- Linear Model) –Evaluate Violations of Assumptions Graphical Analysis of Residuals.](https://reader036.fdocuments.us/reader036/viewer/2022062503/5a4d1ae17f8b9ab05997717e/html5/thumbnails/8.jpg)
Normality & Constant Variance Assumptions
Y
f(e)
X
X1X2
![Page 9: Residual Analysis Purposes –Examine Functional Form (Linear vs. Non- Linear Model) –Evaluate Violations of Assumptions Graphical Analysis of Residuals.](https://reader036.fdocuments.us/reader036/viewer/2022062503/5a4d1ae17f8b9ab05997717e/html5/thumbnails/9.jpg)
A Normal Regression Surface
Y
f(e)
X
X1X2
Every cross-sectional slice of the surface is a normal curve.
![Page 10: Residual Analysis Purposes –Examine Functional Form (Linear vs. Non- Linear Model) –Evaluate Violations of Assumptions Graphical Analysis of Residuals.](https://reader036.fdocuments.us/reader036/viewer/2022062503/5a4d1ae17f8b9ab05997717e/html5/thumbnails/10.jpg)
Analysis of Residuals
A residual is the difference between the actual value of Y and the
predicted value .Y
![Page 11: Residual Analysis Purposes –Examine Functional Form (Linear vs. Non- Linear Model) –Evaluate Violations of Assumptions Graphical Analysis of Residuals.](https://reader036.fdocuments.us/reader036/viewer/2022062503/5a4d1ae17f8b9ab05997717e/html5/thumbnails/11.jpg)
Linear Regression and Correlation Assumptions
• The independent variables and the dependent variable have a linear relationship.
• The dependent variable must be continuous and at least interval-scale.
![Page 12: Residual Analysis Purposes –Examine Functional Form (Linear vs. Non- Linear Model) –Evaluate Violations of Assumptions Graphical Analysis of Residuals.](https://reader036.fdocuments.us/reader036/viewer/2022062503/5a4d1ae17f8b9ab05997717e/html5/thumbnails/12.jpg)
Linear Regression Assumptions • Normality
Y Values Are Normally Distributed with a mean of Zero For Each X. heresiduals ( )are normally distributed with a mean of Zero.
Homoscedasticity (Constant Variance) The variation in the residuals must be the same for all values of Y. The standard deviation of the residuals is the same regardless of the given
value of X.
Independence of Errors The residuals are independent for each value of X The residuals ( ) are independent of each other The size of the error for a particular value of x is not related to the size of
the error for any other value of x
![Page 13: Residual Analysis Purposes –Examine Functional Form (Linear vs. Non- Linear Model) –Evaluate Violations of Assumptions Graphical Analysis of Residuals.](https://reader036.fdocuments.us/reader036/viewer/2022062503/5a4d1ae17f8b9ab05997717e/html5/thumbnails/13.jpg)
Evaluating the Aptness of the Fitted Regression Model
Does the model appear linear?
![Page 14: Residual Analysis Purposes –Examine Functional Form (Linear vs. Non- Linear Model) –Evaluate Violations of Assumptions Graphical Analysis of Residuals.](https://reader036.fdocuments.us/reader036/viewer/2022062503/5a4d1ae17f8b9ab05997717e/html5/thumbnails/14.jpg)
Residual Plot for Linearity(Functional Form)
Aptness of the Fitted Model
Correct Specification
X
e
Add X2 Term
X
e
![Page 15: Residual Analysis Purposes –Examine Functional Form (Linear vs. Non- Linear Model) –Evaluate Violations of Assumptions Graphical Analysis of Residuals.](https://reader036.fdocuments.us/reader036/viewer/2022062503/5a4d1ae17f8b9ab05997717e/html5/thumbnails/15.jpg)
Residual Plots for LinearityResidual Plots for Linearityof the Fitted Modelof the Fitted Model
• Scatter Plot of Y vs. X value• Scatter Plot of residuals vs. X value
![Page 16: Residual Analysis Purposes –Examine Functional Form (Linear vs. Non- Linear Model) –Evaluate Violations of Assumptions Graphical Analysis of Residuals.](https://reader036.fdocuments.us/reader036/viewer/2022062503/5a4d1ae17f8b9ab05997717e/html5/thumbnails/16.jpg)
Using SPSS to Test for Linearity of the Regression Model
• Analyze/Regression/Linear– Dependent - Sales– Independent - Customers– Save
• Predicted Value (Unstandardized or Standardized)• Residual (Unstandardizedor Standardized)
• Graphs/Scatter/Simple• Y-Axis: residual [ res_1 or zre_1 ]• X-Axis: Customer (independent variable)
![Page 17: Residual Analysis Purposes –Examine Functional Form (Linear vs. Non- Linear Model) –Evaluate Violations of Assumptions Graphical Analysis of Residuals.](https://reader036.fdocuments.us/reader036/viewer/2022062503/5a4d1ae17f8b9ab05997717e/html5/thumbnails/17.jpg)
Sales and Customers Problem
.85945
.54359-.00009.31852.10951
-.10343-.60249-.14501.19914.03063
-.72027-.60503
-1.02895-.08175.55129.16327
-.02414.43939.53032
12345678910111213141516171819
Unstandardized Residual
Sales and Customers Problem a
907 11 10.34055 .85945926 11 10.50641 .54359506 7 6.84009 -.00009741 9 8.89148 .31852789 9 9.31049 .10951889 10 10.18343 -.10343874 9 10.05249 -.60249510 7 6.87501 -.14501529 7 7.04086 .19914420 6 6.08937 .03063679 8 8.35027 -.72027872 9 10.03503 -.60503924 9 10.48895 -1.02895607 8 7.72175 -.08175452 7 6.36871 .55129729 9 8.78673 .16327794 9 9.35414 -.02414844 10 9.79061 .43939
1010 12 11.23968 .53032
12345678910111213141516171819
CUSTOMER SALES
Unstandardized Predicted
ValueUnstandardized Residual
![Page 18: Residual Analysis Purposes –Examine Functional Form (Linear vs. Non- Linear Model) –Evaluate Violations of Assumptions Graphical Analysis of Residuals.](https://reader036.fdocuments.us/reader036/viewer/2022062503/5a4d1ae17f8b9ab05997717e/html5/thumbnails/18.jpg)
Scatter Plot of Customer by Sales
CUSTOMER
11001000900800700600500400
SA
LES
12
11
10
9
8
7
6
![Page 19: Residual Analysis Purposes –Examine Functional Form (Linear vs. Non- Linear Model) –Evaluate Violations of Assumptions Graphical Analysis of Residuals.](https://reader036.fdocuments.us/reader036/viewer/2022062503/5a4d1ae17f8b9ab05997717e/html5/thumbnails/19.jpg)
Scatter Plot of Customer by Residuals
CUSTOMER
11001000900800700600500400
Uns
tand
ardi
zed
Res
idua
l1.0
.5
0.0
-.5
-1.0
-1.5
![Page 20: Residual Analysis Purposes –Examine Functional Form (Linear vs. Non- Linear Model) –Evaluate Violations of Assumptions Graphical Analysis of Residuals.](https://reader036.fdocuments.us/reader036/viewer/2022062503/5a4d1ae17f8b9ab05997717e/html5/thumbnails/20.jpg)
Plot of Residuals vs R&D ExpendituresPlot of Residuals vs X Values
RDEXPEND
1614121086420
Res
idua
l
60
40
20
0
-20
-40
ELECTRONIC FIRMS
![Page 21: Residual Analysis Purposes –Examine Functional Form (Linear vs. Non- Linear Model) –Evaluate Violations of Assumptions Graphical Analysis of Residuals.](https://reader036.fdocuments.us/reader036/viewer/2022062503/5a4d1ae17f8b9ab05997717e/html5/thumbnails/21.jpg)
TheLinear Regression Assumptions
1. Normality of residuals (Errors)2. Homoscedasticity (Constant Variance)3. Independence of Residuals (Errors)
Need to verify using residual analysis.
![Page 22: Residual Analysis Purposes –Examine Functional Form (Linear vs. Non- Linear Model) –Evaluate Violations of Assumptions Graphical Analysis of Residuals.](https://reader036.fdocuments.us/reader036/viewer/2022062503/5a4d1ae17f8b9ab05997717e/html5/thumbnails/22.jpg)
Residual Plots for NormalityResidual Plots for Normality• Construct histogram of residuals
– Stem-and-leaf plot– Box-and-whisker plot– Normal probability plot
• Scatter Plot residuals vs. X values– Simple regression
• Scatter Plot residuals vs. Y– Multiple regression
![Page 23: Residual Analysis Purposes –Examine Functional Form (Linear vs. Non- Linear Model) –Evaluate Violations of Assumptions Graphical Analysis of Residuals.](https://reader036.fdocuments.us/reader036/viewer/2022062503/5a4d1ae17f8b9ab05997717e/html5/thumbnails/23.jpg)
Residual Plot 1 for Residual Plot 1 for NormalityNormalityConstruct histogram of residuals
• Nearly symmetric• Centered near or at zero• Shape is approximately normal
RESIDUAL
3.02.01.00.0-1.0-2.0-3.0
10
8
6
4
2
0
Std. Dev = 1.61 Mean = 0.0N = 31.00
![Page 24: Residual Analysis Purposes –Examine Functional Form (Linear vs. Non- Linear Model) –Evaluate Violations of Assumptions Graphical Analysis of Residuals.](https://reader036.fdocuments.us/reader036/viewer/2022062503/5a4d1ae17f8b9ab05997717e/html5/thumbnails/24.jpg)
Using SPSS to Test for NormalityHistogram of Residuals
• Analyze/Regression/Linear– Dependent - Sales– Independent - Customers– Plot/Standardized Residual Plot: Histogram– Save
• Predicted Value (Unstandardized or Standardized)• Residual (Unstandardizedor Standardized)
• Graphs/Histogram– Variable - residual (Unstandardized or Standardedized)
![Page 25: Residual Analysis Purposes –Examine Functional Form (Linear vs. Non- Linear Model) –Evaluate Violations of Assumptions Graphical Analysis of Residuals.](https://reader036.fdocuments.us/reader036/viewer/2022062503/5a4d1ae17f8b9ab05997717e/html5/thumbnails/25.jpg)
Regression Standardized Residual
1.501.00.500.00-.50-1.00-1.50-2.00
Histogram
Dependent Variable: SALESFr
eque
ncy
7
6
5
4
3
2
1
0
Std. Dev = .97
Mean = 0.00
N = 20.00
Histogram of Residuals of Sales and Customer Problemfrom regression output
![Page 26: Residual Analysis Purposes –Examine Functional Form (Linear vs. Non- Linear Model) –Evaluate Violations of Assumptions Graphical Analysis of Residuals.](https://reader036.fdocuments.us/reader036/viewer/2022062503/5a4d1ae17f8b9ab05997717e/html5/thumbnails/26.jpg)
Unstandardized Residual
.75.50.250.00-.25-.50-.75-1.00
7
6
5
4
3
2
1
0
Std. Dev = .49
Mean = 0.00
N = 20.00
Histogram of Residuals of Sales and Customer Problemfrom graph output
![Page 27: Residual Analysis Purposes –Examine Functional Form (Linear vs. Non- Linear Model) –Evaluate Violations of Assumptions Graphical Analysis of Residuals.](https://reader036.fdocuments.us/reader036/viewer/2022062503/5a4d1ae17f8b9ab05997717e/html5/thumbnails/27.jpg)
Residual Plot 2 for Residual Plot 2 for NormalityNormalityPlot residuals vs. X values
• Points should be distributed about the horizontal line at 0
• Otherwise, normality is violated
X
Residuals
0
![Page 28: Residual Analysis Purposes –Examine Functional Form (Linear vs. Non- Linear Model) –Evaluate Violations of Assumptions Graphical Analysis of Residuals.](https://reader036.fdocuments.us/reader036/viewer/2022062503/5a4d1ae17f8b9ab05997717e/html5/thumbnails/28.jpg)
Using SPSS to Test for NormalityScatter Plot
• Simple Regression– Graph/Scatter/Simple
• Y-Axis: residual [ res_1 or zre_1 ]• X-Axis: Customers [independent variable ]
• Multiple Regression– Graph/Scatter/Simple
• Y-Axis: residual [ res_1 or zre_1 ]• X-Axis: predicted Y values
![Page 29: Residual Analysis Purposes –Examine Functional Form (Linear vs. Non- Linear Model) –Evaluate Violations of Assumptions Graphical Analysis of Residuals.](https://reader036.fdocuments.us/reader036/viewer/2022062503/5a4d1ae17f8b9ab05997717e/html5/thumbnails/29.jpg)
Scatter Plot of Customer by Residuals
CUSTOMER
11001000900800700600500400
Uns
tand
ardi
zed
Res
idua
l1.0
.5
0.0
-.5
-1.0
-1.5
![Page 30: Residual Analysis Purposes –Examine Functional Form (Linear vs. Non- Linear Model) –Evaluate Violations of Assumptions Graphical Analysis of Residuals.](https://reader036.fdocuments.us/reader036/viewer/2022062503/5a4d1ae17f8b9ab05997717e/html5/thumbnails/30.jpg)
An accounting standards board investigating the treatment of research and development expenses by the nation’s major electronic firms was interested in the relationship between a firm’s research and development expenditures and its earnings.
The Electronic FirmsThe Electronic Firms
Earnings = 6.840 + 10.671(rdexpend)
![Page 31: Residual Analysis Purposes –Examine Functional Form (Linear vs. Non- Linear Model) –Evaluate Violations of Assumptions Graphical Analysis of Residuals.](https://reader036.fdocuments.us/reader036/viewer/2022062503/5a4d1ae17f8b9ab05997717e/html5/thumbnails/31.jpg)
ELECTRONIC FIRMS
RDEXPEND EARNINGS PRE_1 RES_1 ZPR_1 ZRE_1
15.00 221.00 166.90075 54.09925 1.84527 2.39432 8.50 83.00 97.54224 -14.54224 .48229 -.64361 12.00 147.00 134.88913 12.11087 1.21620 .53600 6.50 69.00 76.20116 -7.20116 .06291 -.31871 4.50 41.00 54.86008 -13.86008 -.35647 -.61342 2.00 26.00 28.18373 -2.18373 -.88070 -.09665 .50 35.00 12.17792 22.82208 -1.19523 1.01006 1.50 40.00 22.84846 17.15154 -.98554 .75909 14.00 125.00 156.23021 -31.23021 1.63558 -1.38218 9.00 97.00 102.87751 -5.87751 .58713 -.26013 7.50 53.00 86.87170 -33.87170 .27260 -1.49909 .50 12.00 12.17792 -.17792 -1.19523 -.00787 2.50 34.00 33.51900 .48100 -.77585 .02129 3.00 48.00 38.85427 9.14573 -.67101 .40477 6.00 64.00 70.86589 -6.86589 -.04194 -.30387
List of Data, Predicted Values and Residuals
Data Predicted Residual Standardized Standardized Value Predicted Value Residual
![Page 32: Residual Analysis Purposes –Examine Functional Form (Linear vs. Non- Linear Model) –Evaluate Violations of Assumptions Graphical Analysis of Residuals.](https://reader036.fdocuments.us/reader036/viewer/2022062503/5a4d1ae17f8b9ab05997717e/html5/thumbnails/32.jpg)
Std. Dev = .96 Mean = 0.00N = 15.00
Regression Standardized Residual
2.502.00
1.501.00
.500.00
-.50-1.00
-1.50
HistogramDependent Variable: EARNINGS
Freq
uenc
y
6543210
ELECTRONIC FIRMS
![Page 33: Residual Analysis Purposes –Examine Functional Form (Linear vs. Non- Linear Model) –Evaluate Violations of Assumptions Graphical Analysis of Residuals.](https://reader036.fdocuments.us/reader036/viewer/2022062503/5a4d1ae17f8b9ab05997717e/html5/thumbnails/33.jpg)
Plot of St. Residuals vs RDexpendPlot of Standardized Residuals vs X Value
RDEXPEND
1614121086420
Stan
dard
ized
Res
idua
l
3
2
1
0
-1-2
ELECTRONIC FIRMS
![Page 34: Residual Analysis Purposes –Examine Functional Form (Linear vs. Non- Linear Model) –Evaluate Violations of Assumptions Graphical Analysis of Residuals.](https://reader036.fdocuments.us/reader036/viewer/2022062503/5a4d1ae17f8b9ab05997717e/html5/thumbnails/34.jpg)
Residual Plot for HomoscedasticityConstant Variance
Correct Specification
X
SR
0
Heteroscedasticity
X
SR
0
Fan-Shaped.Standardized Residuals Used.
![Page 35: Residual Analysis Purposes –Examine Functional Form (Linear vs. Non- Linear Model) –Evaluate Violations of Assumptions Graphical Analysis of Residuals.](https://reader036.fdocuments.us/reader036/viewer/2022062503/5a4d1ae17f8b9ab05997717e/html5/thumbnails/35.jpg)
• Simple Regression– Graphs/Scatter/Simple
• Y-Axis: residual [ res_1 or zre_1 ]• X-Axis: rdexpend [independent variable ]
• Multiple Regression– Graphs/Scatter/Simple
• Y-Axis: residual [ res_1 or zre_1 ]• X-Axis: predicted Y values
Using SPSS to Test for Homoscedasticity of Residuals
![Page 36: Residual Analysis Purposes –Examine Functional Form (Linear vs. Non- Linear Model) –Evaluate Violations of Assumptions Graphical Analysis of Residuals.](https://reader036.fdocuments.us/reader036/viewer/2022062503/5a4d1ae17f8b9ab05997717e/html5/thumbnails/36.jpg)
Test for Homoscedasticity
Plot of Residuals vs Number
NUMBER
6543210
Res
idua
l
1.5
1.0
.5
0.0
-.5
-1.0
-1.5
DUNTON’S WORLD OF SOUND
![Page 37: Residual Analysis Purposes –Examine Functional Form (Linear vs. Non- Linear Model) –Evaluate Violations of Assumptions Graphical Analysis of Residuals.](https://reader036.fdocuments.us/reader036/viewer/2022062503/5a4d1ae17f8b9ab05997717e/html5/thumbnails/37.jpg)
Plot of Residuals vs R&D ExpendituresPlot of Residuals vs X Values
RDEXPEND
1614121086420
Res
idua
l
60
40
20
0
-20
-40
Test for Homoscedasticity
ELECTRONIC FIRMS
![Page 38: Residual Analysis Purposes –Examine Functional Form (Linear vs. Non- Linear Model) –Evaluate Violations of Assumptions Graphical Analysis of Residuals.](https://reader036.fdocuments.us/reader036/viewer/2022062503/5a4d1ae17f8b9ab05997717e/html5/thumbnails/38.jpg)
Scatter Plot of Customer by Residuals
CUSTOMER
11001000900800700600500400
Uns
tand
ardi
zed
Res
idua
l1.0
.5
0.0
-.5
-1.0
-1.5
![Page 39: Residual Analysis Purposes –Examine Functional Form (Linear vs. Non- Linear Model) –Evaluate Violations of Assumptions Graphical Analysis of Residuals.](https://reader036.fdocuments.us/reader036/viewer/2022062503/5a4d1ae17f8b9ab05997717e/html5/thumbnails/39.jpg)
Residual Plot for Independence
Correct Specification
X
SR
Not Independent
X
SR
Plots Reflect Sequence Data Were Collected.
![Page 40: Residual Analysis Purposes –Examine Functional Form (Linear vs. Non- Linear Model) –Evaluate Violations of Assumptions Graphical Analysis of Residuals.](https://reader036.fdocuments.us/reader036/viewer/2022062503/5a4d1ae17f8b9ab05997717e/html5/thumbnails/40.jpg)
Two Types of Autocorrelation
• Positive Autocorrelation: successive terms in time series are directly related
• Negative Autocorrelation: successive terms are inversely related
![Page 41: Residual Analysis Purposes –Examine Functional Form (Linear vs. Non- Linear Model) –Evaluate Violations of Assumptions Graphical Analysis of Residuals.](https://reader036.fdocuments.us/reader036/viewer/2022062503/5a4d1ae17f8b9ab05997717e/html5/thumbnails/41.jpg)
0
20
-20
0 4 8 12 16 20
Residualy - y
Time Period, t
Positive autocorrelation:Residuals tend to be followedby residuals with the same sign
![Page 42: Residual Analysis Purposes –Examine Functional Form (Linear vs. Non- Linear Model) –Evaluate Violations of Assumptions Graphical Analysis of Residuals.](https://reader036.fdocuments.us/reader036/viewer/2022062503/5a4d1ae17f8b9ab05997717e/html5/thumbnails/42.jpg)
0
20
-20
0 4 8 12 16 20
Residualy - y
Time Period, t
Negative Autocorrelation:Residuals tend to change signsfrom one period to the next
![Page 43: Residual Analysis Purposes –Examine Functional Form (Linear vs. Non- Linear Model) –Evaluate Violations of Assumptions Graphical Analysis of Residuals.](https://reader036.fdocuments.us/reader036/viewer/2022062503/5a4d1ae17f8b9ab05997717e/html5/thumbnails/43.jpg)
Problems with autocorrelated time-series data
• sy.x and sb are biased downwards• Invalid probability statements about
regression equation and slopes• F and t tests won’t be valid• May imply that cycles exist• May induce a falsely high or low agreement
between 2 variables
![Page 44: Residual Analysis Purposes –Examine Functional Form (Linear vs. Non- Linear Model) –Evaluate Violations of Assumptions Graphical Analysis of Residuals.](https://reader036.fdocuments.us/reader036/viewer/2022062503/5a4d1ae17f8b9ab05997717e/html5/thumbnails/44.jpg)
Using SPSS to Test for Independence of Errors
• Graphs/Sequence– Variables: residual (res_1)
• Durbin-Watson Statistic
![Page 45: Residual Analysis Purposes –Examine Functional Form (Linear vs. Non- Linear Model) –Evaluate Violations of Assumptions Graphical Analysis of Residuals.](https://reader036.fdocuments.us/reader036/viewer/2022062503/5a4d1ae17f8b9ab05997717e/html5/thumbnails/45.jpg)
Time Sequence of Residuals
Sequence number
7654321
Res
idua
l
1.5
1.0
.5
0.0
-.5
-1.0
-1.5
DUNTON’S WORLD OF SOUND
![Page 46: Residual Analysis Purposes –Examine Functional Form (Linear vs. Non- Linear Model) –Evaluate Violations of Assumptions Graphical Analysis of Residuals.](https://reader036.fdocuments.us/reader036/viewer/2022062503/5a4d1ae17f8b9ab05997717e/html5/thumbnails/46.jpg)
Sequence number
151413121110987654321
Time Sequence Plot of ResidualsRe
sidu
al
60
40
20
0
-20
-40
ELECTRONIC FIRMS
![Page 47: Residual Analysis Purposes –Examine Functional Form (Linear vs. Non- Linear Model) –Evaluate Violations of Assumptions Graphical Analysis of Residuals.](https://reader036.fdocuments.us/reader036/viewer/2022062503/5a4d1ae17f8b9ab05997717e/html5/thumbnails/47.jpg)
794 9799 8837 7855 9845 10844 10863 11875 11880 12905 13886 12843 10904 12950 12841 10
Customers Sales($000)
Customers and sales for period of 15 consecutive weeks.
![Page 48: Residual Analysis Purposes –Examine Functional Form (Linear vs. Non- Linear Model) –Evaluate Violations of Assumptions Graphical Analysis of Residuals.](https://reader036.fdocuments.us/reader036/viewer/2022062503/5a4d1ae17f8b9ab05997717e/html5/thumbnails/48.jpg)
Residuals over Time
Time
151413121110987654321
Uns
tand
ardi
zed
Res
idua
l2
1
0
-1
-2
-3
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Durbin-Watson Procedure• Used to Detect Autocorrelation
– Residuals in One Time Period Are Related to Residuals in Another Period
– Violation of Independence Assumption• Durbin-Watson Test Statistic
D(e e
e
i ii
n
ii
n
12
2
2
1
)
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H0 : No positive autocorrelation exists (residuals are random)H1 : Positive autocorrelation exists
Accept Ho if d> du
Reject Ho if d < dL
Inconclusive if dL < d < du
d =
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Testing for Positive Autocorrelation
There is positiveautocorrelation
The test isinconclusive
There is no evidence of autocorrelation
0 dL du2 4
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Rule of Thumb
• Positive autocorrelation - D will approach 0• No autocorrelation - D will be close to 2• Negative autocorrelation - D is greater than 2
and may approach a maximum of 4
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Using SPSS with Autocorrelation
• Analyze/Regression/Linear• Dependent; Independent• Statistics/Durbin-Watson (use only time series
data)
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794 9799 8837 7855 9845 10844 10863 11875 11880 12905 13886 12843 10904 12950 12841 10
Customers Sales($000)
Customers and sales for period of 15 consecutive weeks.
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Residuals over Time
Time
151413121110987654321
Uns
tand
ardi
zed
Res
idua
l2
1
0
-1
-2
-3
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Model Summaryb
.811a .657 .631 .94 .883Model1
R R SquareAdjustedR Square
Std. Error ofthe Estimate
Durbin-Watson
Predictors: (Constant), CUSTOMERa.
Dependent Variable: SALESb.
Durbin-Watson.883
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Using SPSS with Autocorrelation
• Analyze/Regression/Linear• Dependent; Independent• Statistics/ Durbin-Watson (use only time series data) • If DW indicates autocorrelation, then …
– Analyze/Time Series/Autoregression– Cochrane-Orcutt– OK
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Solutions for autocorrelation• Use Final Parameters under Cochrane-Orcutt• Changes in the dependent and independent variables -
first differences• Transform the variables• Include an independent variable that measures the time of
the observation• Use lagged variables (once lagged value of dependent
variable is introduced as independent variable, Durbon-Watson test is not valid