Autocorrelation

Business Statistics: A Decision-Making Approach, 7e 2008 Prentice-Hall, Inc.

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

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

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.

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.

No pattern in residuals No autocorrelation

No pattern in residuals at all: this is what we would like to see

Positive Autocorrelation

Positive Autocorrelation is indicated by a cyclical residual plot over time.

Negative Autocorrelation

Negative autocorrelation is indicated by an alternating pattern where the residualscross the time axis more frequently than if they were distributed randomly

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

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

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

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)

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

SPSS