autocorrelation final
DESCRIPTION
AutocorrelationTRANSCRIPT
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Autocorrelation
Business Statistics: A Decision-Making Approach, 7e 2008 Prentice-Hall, Inc.
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AutocorrelationAssumption of OLS regression is that error terms are independent and normally distributed, with a mean zero and a constant variance
If this assumption violate serial correlation is indicated
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AutocorrelationAutocorrelation is correlation of the error terms (residuals) over time
(continued)Violates the regression assumption that residuals are random and independentHere, residuals show a cyclic pattern, not random
Chart1
-1.9047619048
8.5238095238
-11.0476190476
-0.619047619
9.8095238095
-4.7619047619
Time (t)
Residuals
Time (t) Residual Plot
Sheet1
YearCPIIndex (1983 = base year)
1975161.254.0
1976170.557.1
1977181.560.8
1978195.465.5
1979217.472.9
1980246.882.7
1981272.491.3
1982289.196.9
1983298.4100.0
1984311.1104.3
1985322.2108.0
1986328.4110.1
1987340.4114.1
1988354.3118.7
1989371.3124.4
1990391.4131.2
1991408.0136.7
1992420.3140.9
1993432.7145.0
1994444.0148.8
1995456.5153.0
1996469.9157.5
1997480.8161.1
1998488.3163.6
1999499.0167.2
2000515.8172.9
2001530.4177.7
2002538.8180.6
Sheet2
YearNumber of OrdersIndex
(base year = 1997)
199527285.0
199628890.0
199729592.2
199831197.2
1999322100.6
2000320100.0
2001348108.8
2002366114.4
2003384120.0
Sheet3
Automobile Expenses:
Monthly Amounts ($):
YearLease paymentGasMaintenanceTotalIndex (2001=100)
20012604540345100.0
20022806040380110.1
20033055545405117.4
20043105050410118.8
SLR
Regression Analysis
Regression Statistics
Multiple R0.9135891753
R Square0.8346451813
Adjusted R Square0.7933064766
Standard Error8.9109349624
Observations6
ANOVA
dfSSMSFSignificance F
Regression11603.21428571431603.214285714320.19040479760.0108776384
Residual4317.61904761979.4047619048
Total51920.8333333333
CoefficientsStandard Errort StatP-valueLower 95%Upper 95%
Intercept12.33333333338.29562898871.48672672680.2112881343-10.699072872435.365739539
Time (t)9.57142857142.13012088064.49337343180.01087763843.657252629315.4856045135
RESIDUAL OUTPUT
ObservationPredicted Sales (y)Residuals
121.9047619048-1.9047619048
231.47619047628.5238095238
341.0476190476-11.0476190476
450.619047619-0.619047619
560.19047619059.8095238095
669.7619047619-4.7619047619
SLR
1
2
3
4
5
6
Time (t)
Residuals
Time (t) Residual Plot
Sheet4
YearTime (t)Sales (y)
1999120
2000240
2001330
2002450
2003570
2004665
Sheet4
Sales (y)
Year
sales
Sales
Sheet5
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More common in time series dataif we are dealing with time series data, for the observations in such data follow a natural ordering over time so that successive observations are likely to exhibit intercorrela- tions, especially if the time interval between successive observations is short, such as a day, a week, or a month rather than a year. If you observe stock price indexes, such as the Dow Jones or S&P 500 over successive days, it is not unusual to nd that these indexes move up or down for several days in succession. Obviously, in situations like this, the assumption of no auto, or serial, correlation in the error terms will be violated.
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Autocorrelation can arise for several reasonsinertia or sluggishness of economic time seriesspecication bias resulting from excluding important variables from the modelusing incorrect functional formdata transformation.
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No pattern in residuals No autocorrelation
No pattern in residuals at all: this is what we would like to see
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Positive Autocorrelation
Positive Autocorrelation is indicated by a cyclical residual plot over time.
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Negative Autocorrelation
Negative autocorrelation is indicated by an alternating pattern where the residualscross the time axis more frequently than if they were distributed randomly
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problemsMakes the estimates of the standard errors smaller than the true standard error. This means that t ratio calculated will be overstatedR square and F statistic unreliable
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causeExistence of long term cycles and trends in business and economic data.this leads to positive serial correlationAlso caused by mis specification.leaving one or more important variable or not including a non linear term when needed
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How to removeUse first differences rather than actual valuesSpecify the model by adding variablesIntroduce the square of an existing causal variable as an independent variableIntroduce a lag on dependent variable as an independent variableIntroduce a lag of the dependent variable as an independent variable
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Testing for AutocorrelationThe Durbin-Watson Statistic is used to test for autocorrelationH0: = 0 (residuals are not correlated)HA: > 0 (positive autocorrelation is present)Durbin-Watson test statistic:d = Durbin-Watson test statisticei = = residual at time tn = number of time periods in the time series(The test for positive autocorrelation is the most common test in business applications)
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Measuring autocorrelation in SPSSAutocorrelation is measured by Durbin Watson StatisticThe value of Durbin Watson statistic ranges from 0 to 4If it is between 2 and 4-negative autocorrelationIf it is between 0 and 2-positive autocorrelation
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SPSS