Autocorrelation

1
Autocorrelation

• Hill et al Chapter 12

2
The nature of the problem

• Time series data
• Observations follow a natural ordering through
  time.
• Autocorrelation
• The error term contains a carryover from previous
  shocks.
• Related to, or correlated with, the effects of
  the earlier shocks.

3
Violation of assumption MR4
4
Area response for sugar cane
5
Least squares estimation
6
First order autoregressive AR(1) errors
7
Properties of an AR(1) error
Assumption
Properties
8
Consequences of Autocorrelation

• The least squares estimator is still a linear
  unbiased estimator, but it is no longer best.
• The formulas for the standard errors usually
  computed for the least squares estimator are no
  longer correct
• Hence confidence intervals and hypothesis tests
  that use these standard errors may be misleading.

9
Transforming the model
10
Transforming the first observation

11
Implementing GLS
12
Sugar cane area continued

1 0.93970 -2.5868 -2.4308 3.3673 3.1642
2 0.65799 -2.1637 -1.2790 4.2627 3.1110
3 0.65799 -2.2919 -1.5519 3.7677 2.2798
4 0.65799 -2.2045 -1.4206 4.4998 3.2215
13
The Durbin-Watson test
14
Critical values for D-W

• The distribution of d under the null depends on
  the values of the explanatory variables.
• Critical values cannot be tabulated.
• Use software to compute appropriate p-value.
• Use the bounds test.
• Define two new stats which do not have a
  distribution dependent on the data.
• dL lt d lt du

15
Critical values for the bounds test
Sugar Cane Example
16
A lagrange multiplier test
17
Points to note in testing for autocorrelation

• In the LM test, the estimated residual for t1 is
  missing. Either omit the first observation or
  make e00.
• D-W test is exact in finite samples, LM is a
  large sample test.
• D-W is not valid when one of the variables is a
  lagged dependent variable.
• The LM test can be used for higher order forms of
  autocorrelation.

18
Prediction with AR(1) errors