Detection of Auto-Correlation
There are several methods to detect the
autocorrelation in a regression model. Below are a few of them:
1.
Graphic Method
A crude way to detect the autocorrelation. Construct a regression line in
the first step and plot et = Yt - estimated Y on the regression line. The absence of any
systematic pattern leads to the conclusion that there is no autocorrelation in
the model, but the formation of any systematic pattern indicates the presence
of autocorrelation in the model.
In the second stage, plot et against et-1 into a two-dimensional diagram. If most of the
points (et, et-1) fall in Quadrants I and III, the
autocorrelation will be positive, and if most of the points (et, et-1) fall in Quadrants II and IV, the
autocorrelation will be negative.
2.
The Durbin Watson – d Test
The Durbin Watson d-test
is commonly used for detection of autocorrelation. This test is applicable
when the sample size is small (n < 30) and the error term follows the AR (1) scheme.
The Durbin Watson-d-test is based on the following assumptions:
i. The regression model
has an intercept.
ii. The predictor variable
is free from error (non-stochastic).
iii. The noise term is
generated by a first-order auto-regressive scheme.
iv. The regression model
does not have a lagged value of the response variable.
v. There will be no
missing observation in the data.
The Durbin Watson-d-test statistic is given by:
Decision Rules for
Applying the Durbin Watson-d-test:
Limitations of Durbin Watson d - test
i. If d falls in the inconclusive zone, then
no conclusive inference can be drawn.
ii. The D-W test is not applicable when the intercept term is absent in the model.
iii. The test is not valid when lagged
dependent variables appear as explanatory variables.
Practice Question
A researcher in education establishes a linear relationship between the enrolment of students and each succeeding semester.
The researcher gathers the information in the following ways:
|
Semester |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
|
No. of Students Enrolled |
60 |
70 |
75 |
62 |
72 |
68 |
54 |
82 |
Draw the regression model of enrolment students on semester and
check the existence of autocorrelation. If the auto correlation exists, check
whether it is positive or negative.
Solution:
The OLS estimates are given by:
The estimated regression model on enrolment of students on
semester is given by:
Practice Question
Estimate the demand and test the existence of an autocorrelation.
Solution: Estimate the
parameters of the demand model by O L S.
The estimated demand model:
3.
Durbin Watson h test
(Detection
of auto-correlation with a lagged dependent variable)
When
the lag value of a response variable is used as a regressor in a regression
model. The Durbin-Watson d statistic is not for the detection of
autocorrelation.
The Durbin h statistic, which is also obtained from the residuals, can be used in this situation to conduct a Durbin h test. Here is a description of the h statistic:
Note that n will usually be one less than the number of
observations in the sample because the first observation is lost when the
equation is fitted. There are various ways in which one might estimate a sample estimate, but since this test is valid only for large
samples, it does not matter which we use. The most convenient is to take
advantage of the large sample relationship between d and parameter.
The estimate of the variance of the coefficient of the lagged
dependent variable is obtained by squaring its standard error. In large samples,
h is distributed as N (0, 1). We test the
hypothesis as:
Practice Question
The economic model of consumption at time “t” is given by:
Where d = 1.529 and n = 120. Test the existence of autocorrelation at the 5% level of
significance.
Solution: The model includes the lag value of the dependent variable as an independent variable, so the Durbin-Watson d test is not applicable. Using Durbin
Watson h test;
The above result indicates that the autocorrelation presents in the model.
4.
The Geary Runs Test
To check the existence of autocorrelation, we use the runs test. A run means the subsequence of the same sign preceded and followed by different signs or no sign at all. The test statistic is denoted by nr (number of runs) of et.
Where n is the total number of observations.
n1 is the number of positive symbols.
n2 is the number of negative symbols.
Number
of too many runs indicating negative autocorrelation, while a few runs
indicating positive autocorrelation.
Here
we test the hypothesis as:
5.
The Chi Squares Test of Independence of Residuals
To
check the existence of autocorrelation, we can use the chi square test of independence.
First,
we construct a 2 x 3 contingency table as:
The test statistic is given by:
6. Breusch-Godfrey Test
(Lagrange multiplier
test for autocorrelation)
The Breusch-Godfrey test is performed through the auxiliary
regression of the residuals on independent variables (X's) and lagged
residuals. This test is applicable when
i. The regression model includes the lagged value of the dependent variable as a regressor.
ii. higher-order auto-regressive schemes. AR (p)
Consider the model:
Then the Breusch-Godfrey test is used to delete high-order autoregression
and can be carried out as follows:
Step 1: Run the OLS
regression to calculate an estimate of the model
Step 2: Regress estimated residuals on lagged values up to lag p and all the original RHS X variables
Step 3: Test the null hypothesis as:
Step 4: Either compute the F test for the joint significance of the residuals and if
Or test the null hypothesis as, If the sample size is sufficiently large, then
- Read More: Stochastic Regression
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