Chapter 9: Serial Correlation
In this chapter:
1. Creating a residual series from a regression model
2. Plotting the error term to detect serial correlation (UE, pp. 313-315)
3. Using regression to estimate ρ, the first order serial correlation coefficient (UE,
Equation 9.1, pp. 311-312)
4. Viewing the Durbin-Watson d statistic in the EViews Estimation Output window (UE
9.3)
5. Estimating generalized least squares using the AR(1) method (UE 9.4.2)
6. Estimating generalized least squares (GLS) equations using the Cochrane-Orcutt
method (UE 9.4.2)
7. Exercise
Serial correlation analysis involves an examination of the error term. The demand for
chicken model specified in UE, Equation 6.8, p. 166, will be used to demonstrate most of
the procedures reviewed in this chapter.
Creating a residual series from a regression model:
Follow these steps to estimate the demand for chicken model (UE, Equation 6.8, p. 166),
save the results in an equation named EQ01, make a residual series named E, and save
changes to the workfile:
Step 1. Open the EViews workfile named Chick6.wf1.
Step 2. Select Objects/New Object/Equation on the workfile menu bar and enter Y C PC PB
YD in the Equation Specification: window, and click OK.
Step 3. Select Name on the equation window menu bar, enter EQ01 in the Name to identify
object: window, and click OK.
Step 4. To create a new series for the residuals
(errors) for EQ01, select Procs/Make
Residual Series on the equation window menu
bar and the graphic on the right appears. Enter
E in the Name for residual series: window,
click OK, and a spreadsheet view of the
residual series will be displayed in a new
window.
Step 5. Select Save on the workfile menu bar to
save your changes.
Plotting of the error term to detect serial correlation (UE, pp. 313-315):
Complete the section entitled Creating a residual series from a regression model before
attempting this section (i.e., Equation EQ01 and series E should already be present in the
workfile). Follow these steps to view a residuals graph in EViews:
Step 1. Open EQ01, by double clicking the icon in the workfile window.
Step 2. Select View/Actual, Fitted, Residual/Residual Graph on the equation window menu
bar to reveal the graph on the left below. Note the residual series exhibits a pattern akin to
the graphs displayed in UE, Figure 9.1, p. 313. Thus, graphical analysis indicates positive
serial correlation. Steps 3 and 4 below show how to generate a time series plot of the
same residual series E.
Step 3. Open the residual series named E in a new window by double clicking the series icon
in the workfile window to open the residual series from EQ01 in a new window.
Step 4. Select View/Line Graph to reveal a time series graph of the residuals shown below.
Using regression to estimate ρ, the first order serial correlation coefficient (UE,
Equation 9.1, pp. 311-312):1
Complete the section entitled Creating a residual series from a regression model before
attempting this section (i.e., Equation EQ01 and series E should already be present in the
workfile). Follow the steps below to estimate the first order serial correlation coefficient
and test for possible first order serial correlation:
Step 1. Open the EViews workfile named Chick6.wf1.
Step 2. Select Objects/New Object/Equation on the workfile menu bar, enter E C E(-1) in
the Equation Specification: window, and click OK to reveal the regression output shown
in the graphic below. Rho (ρ) is used to symbolize the coefficient on E(-1) and it
represents the first-order autocorrelation coefficient in this regression. In this case, the
value of ρ is positive and significant at the 1% level (t-statistic = 3.69 and Prob value =
0.0006). It is important to note that this is not a test of serial correlation, but the value of
ρ is related to the value of the Durbin-Watson d statistic discussed in the next section.2
Step 3. Select Name on the equation menu bar, enter EQ02 in the Name to identify object:
window, and click OK.
Step 4. Select Save on the workfile menu bar to save your changes.
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To test for possible second order serial correlation, regress the residuals against its value lagged one
period and two periods by entering E C E(-1) E(-2) in the Equation Specification: window, and click OK.
To detect seasonal serial correlation in a quarterly model, regress the residuals against its value lagged four
periods enter E C E(-4) in the Equation Specification: window, and click OK. Similarly, to detect seasonal
serial correlation in a monthly model, regress the residuals against its value lagged twelve periods enter E C
E(-12) in the Equation Specification: window, and click OK.
2
The Durbin-Watson d statistic is approximately equal to 2(1-ρ).
Viewing the Durbin-Watson d statistic in the EViews Estimation Output window
(UE 9.3):
Complete the section entitled Creating a residual series from a regression model before
attempting this section (i.e., Equation EQ01 should already be present in the workfile).
Follow these steps to view the Durbin-Watson d test for EQ01:
Step 1. Open EQ01 by double clicking the icon in the workfile window.
Step 2. Select View/Estimation Output on the EQ01 menu bar to reveal the regression output
shown below. The Durbin-Watson statistic is highlighted in yellow and boxed in red.3
Step 3. Use the Sample size printed after Included observations: (i.e., 44) and the number of
explanatory variables listed in the Variable column (i.e., 3) and follow the instructions in
UE, p. 612 to find the upper and lower critical d value in UE, Tables B-4, B-5 or B-6, pp.
613 - 615).
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The Durbin-Watson statistic is a test for first-order serial correlation. More formally, the DW statistic
measures the linear association between adjacent residuals from a regression model. The Durbin-Watson is
a test of the hypothesis ρ=0 in the specification:
εt = ρεt-1 + υt.
If there is no serial correlation, the DW statistic will be around 2. The DW statistic will fall below 2 if there
is positive serial correlation (in the worst case, it will be near zero). If there is negative correlation, the
statistic will lie somewhere between 2 and 4. Positive serial correlation is the most commonly observed
form. As a rule of thumb, with 50 or more observations and only a few independent variables, a DW
statistic below about 1.5 is a strong indication of positive first order serial correlation.
Estimating generalized least squares (GLS) equations using the AR(1) method (UE
9.4.2):
Follow these steps to estimate the chicken demand model using the AR(1) method of
GLS equation estimation.
Step 1. Open the EViews workfile named Chick6.wf1.
Step 2. Select Objects/New Object/Equation on the workfile menu bar, enter Y C PC PB
YD AR(1) in the Equation Specification: window, and click OK to reveal the output
below. EViews automatically adjusts your sample to account for the lagged data used in
estimation, estimates the model, and reports the adjusted sample along with the remainder
of the estimation output.
The estimated coefficients, coefficient standard errors, and t-statistics may be interpreted
in the usual manner. The estimated coefficient on the AR(1) variable is the serial
correlation coefficient of the unconditional residuals.4
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Unconditional residuals are the errors that you would observe if you made a prediction of the value of
using contemporaneous information, but ignoring the information contained in the lagged residual. For AR
models estimated with EViews, the residual-based regression statistics—such as the, the standard error of
regression, and the Durbin-Watson statistic— reported by EViews are based on the one-period-ahead
forecast errors.
The most widely discussed approaches for estimating AR models are the Cochrane-Orcutt, Prais-
Winsten, Hatanaka, and Hildreth-Lu procedures. These are multi-step approaches designed so that
estimation can be performed using standard linear regression. EViews estimates AR models using
nonlinear regression techniques. This approach has the advantage of being easy to understand, generally
applicable, and easily extended to nonlinear specifications and models that contain endogenous right-hand
side variables.
Estimating generalized least squares (GLS) equations using the Cochrane-Orcutt
method (UE 9.4.2):
The Cochrane-Orcutt method is a multi-step procedure that requires re-estimation until
the value for the estimated first order serial correlation coefficient converges. Follow
these steps to use the Cochrane-Orcutt method to estimate the CIA's "high" estimate of
Soviet defense expenditures (i.e., this is UE, Exercise 14, Equation 9.28, p. 342).
Step 1. Open the EViews workfile named Defend9.wf1.
Step 2. Follow the steps in Creating a residual series from a regression model to estimate the
OLS equation LOG(SDH) C LOG(USD) LOG(SY) LOG(SP), name it EQ01, and create a
residual series for EQ01 named E.
Step 3. Estimate ρ, and name it EQ02 .
Step 4. To estimate the generalized differenced form of UE, Equation 9.28, select
Objects/New Object/Equation on the workfile menu bar, enter EQ03 in the Name to
identify object: window, and enter LOG(SDH)-EQ02.@COEFS(2)*LOG(SDH(-1)) C
LOG(USD)-EQ02.@COEFS(2)*LOG(USD(-1)) LOG(SY)-EQ02.@COEFS(2)*LOG(SY(-
1)) LOG(SP)-EQ02.@COEFS(2)*LOG(SP(-1)) in the Equation Specification: window.
The specification should appear as in the figure below.5 Click OK to view the EViews
OLS output. The variable names are truncated in the EViews regression output table
because they don't fit in the variable name cell. Nonetheless, the regression is correct. 6
Step 5. To calculate the new residual series, enter the following formula in the command
window: series E = LOG(SDH)-(EQ03.@COEFS(1) + EQ03.@COEFS(2)*LOG(USD)
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Statistical output for previously saved equations can be recalled by typing the equation name followed by
a period and the reference to the specific output desired. In this case, the value for ρ from EQ02 (recall that
ρ was coefficient # 2 on the E(-1) term) can be recalled with the command EQ02.@coefs(2). The
expression EQ02.@coefs(2) can be used for ρ in the Equation Specification: window.
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The equation can be viewed by selecting View/Representations on the equation menu bar. The equation
should read: LOG(SDH)-EQ02.@COEFS(2)*LOG(SDH(-1)) = 0.1506791883 +
0.107961186*(LOG(USD)-EQ02.@COEFS(2)*LOG(USD(-1))) + 0.1368904004*(LOG(SY)-
EQ02.@COEFS(2)*LOG(SY(-1))) - 0.000837025419*(LOG(SP)-EQ02.@COEFS(2)*LOG(SP(-1))). Select
View/Estimation Output on the group window menu bar to restore the estimation output view for EQ01.
+ EQ03.@COEFS(3)*LOG(SY) + EQ03.@COEFS(4)*LOG(SP)), and press Enter. The
phrase "E successfully computed" should appear in the lower left of your screen.
Step 6. Re-run EQ02, EQ03 and the series E equation7 in Step 6 sequentially until the
estimated ρ (i.e., the coefficient on the E(-1) term from EQ02) does not change by more
that a pre-selected value such as 0.001. After 11 iterations the value for ρ converged (i.e.,
ρ changed from 0.957566 to 0.95758 between the 10th and 11th iteration).
Step 7. Convert the constant from the final version of EQ03 by typing the following formula
in the command window: scalar BETA0=EQ03.@COEFS(1)/(1-EQ02.@COEFS(2)),
and press Enter. Double click the icon in the workfile window and read the
value for the estimated constant in the lower left of the screen.
The final equation is LOG(SDH) = 3.552082480728 + 0.107961186*(LOG(USD)) +
0.1368904004*(LOG(SY)) - 0.000837025419*(LOG(SP)). Note that this is the same
equation reported in UE, Exercise 14, Equation 9.28, p. 342.
Exercise:
15. Follow the steps explained in the Estimating generalized least squares (GLS)
equations using the Cochrane-Orcutt method section, using SDL as the dependent
variable instead of SDH.
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You can re-run an equation by opening the equation in a window, selecting Estimate on the equation
menu bar, and clicking OK. You can re-run the series e equation by clicking the cursor anywhere on the
equation in the command window and hitting Enter on the keyboard.
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The number 3.55208248072 was computed in Step 7 above.
