Serial Correlation
(Section 6)
I.
What the problem is
- Serial correlation is most commonly a problem in time series and
panel data (but can also occur in cross sectional data)
- Sometimes serial correlation is called autocorrelation
Serial Correlation:
Corr (ε r , ε s | X 1... X k ) ≠ 0
for some r ≠ s
Conditional on x’s,
errors of two different observations (two different years in time
series) are correlated
NO Serial Correlation:
Corr (ε r , ε s | X 1 ... X k ) = 0
for all r ≠ s
Conditional on x’s,
errors of two different observations (two different years in time
series) are uncorrelated
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A. Pure serial correlation
1. First order serial correlation in time series data (stochastic error
term follows an AR(1) process)
AR(1) process (Markov scheme):
ε t = ρε t −1 + ut
- What if ρ = 0 ?
- What if ρ → 1 ?
- What if ρ > 1 ?
- What if ρ is positive? (positive serial correlation)
Picture of positive serial correlation:
2
What if ρ is negative? (negative serial correlation)
Picture of negative serial correlation:
Picture of stochastic error with no serial correlation:
2. Other order serial correlation
(a) Seasonally serial correlation
(b) Second order serial correlation (error term is an AR(2) process)
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3. Serial Correlation in panel data
Correlation between errors of a particular individual will probably
be correlated
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B. Impure serial correlation
Impure serial correlation- serial correlation caused by a specification
error (such as an omitted variable or incorrect functional form)
Ex
True equation:
Yt = β 0 + β1 X 1t + β 2 X 2t + ε t
Suppose we accidentally omit X 2 ?
Yt = β 0 + β1 X 1t + ε t*
ε t* = β 2 X 2t + ε t
ε t* will tend to be serial correlation if
1. X 2 is serially correlated (
2. size of ε is small compared to the size of β 2 X 2
) &
Ex (from the book)
Ft = β 0 + β1 RPt + β 2Ydt + β 3 Dt + ε t
What happens if disposable income is omitted?
1. Why might the estimated coefficients for RP & D be bias?
2. Error term included left out disposable income effect
Disposable income probability follows a serially correlated pattern
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Why?
Some more algebra:
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C. Does putting a lagged value of our dependent variable in our
independent variables cause bias and serial correlation?
i.e.
- Most textbooks would say this would cause bias & serial
correlation, but this is not quite right.
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II.
Consequences of serial correlation
Three major consequences of serial correlation:
1. Pure serial correlation does not cause bias in the coefficients’
estimates
2. Serial correlation causes OLS estimates not to be the minimum
variance estimates (of all the linear unbiased estimators)
( )
3. Serial correlation causes the OLS estimates of SE βˆ j ’s to be bias,
leads to unreliable hypothesis testing
- Does not make it βˆ j biased but standard errors will typically
increase because of serial correlation
- OLS is more likely to misestimate β
Note: It very rarely makes sense to talk about a bias in R 2 (or
adjusted R 2 ) caused by serial correlation
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III. Tests for detecting serial correlation
A. A t-test for AR(1) serial correlation with strictly exogenous
regressors
Must assume:
Hypothesis we will test:
How do we test this?
Why can’t we regress
ε t = ρε t −1 + ut
and use t-test on ρ ?
t = 2,...T
What could we do instead?
If we get a small p-value what does this mean?
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B. Durbin-Watson test
Must meet the following assumptions to use the Durbin-Watson test:
1. Regression includes an intercept
2. Serial correlation is AR(1)
3. The regression does not included a lagged dependent variables as
an independent variable
T
∑ (et − et −1 )
DW = t = 2
2
T
∑ (et )2
t =1
DW =0 : Extreme positive serial correlation
DW ≈ 2 : No serial correlation
DW =4 : Extreme negative serial correlation
- Weaknesses of Durbin-Watson test
• Can have inconclusive results
• Depends on a full set of OLS assumptions
- How to perform a Durbin-Watson test
• Run regression → Get OLS residual → Calculate DW stat
• Determine sample size and use tables, in our book (B-4, B-5,
B-6)
Decision rule for
DW < d L
DW > dU
d L ≤ DW ≤ dU
HO : ρ ≤ 0 :
Reject H O
Do not reject H O
Inconclusive
• Decision rule for
If DW < d L
If DW > 4 − dU
If 4 − dU ≤ DW ≤ dU
Otherwise
HO : ρ = 0 :
Reject H O
Reject H O
Do not reject H O
Inclusive
•
If
If
If
- SPSS gives a DW statistic
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C. Other tests
- There is a DW test for AR(2) serial correlation
ε t = ρ1ε t −1 + ρ 2ε t − 2 + ut
H O : ρ1 = 0 ρ 2 = 0
F-Test for joint significance
- There is also a Lagrange multiplier test
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IV. What to do about serial correlation
- What do you do if your Durbin-Watson shows you have serial
correlation
A. First check to see if it is impure serial correlation
- Check specification of your equation
1. Is the functional form correct?
2. Are there omitted variables?
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B. For pure serial correlation
1. Use generalized least squares (GLS)
- GLS starts with an equation that does not meet CLR assumptions
(In this case due to serial correlation) & transform it into one that
does.
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Ways to estimate GLS equations:
a) Cochrane-Orcutt Method (two step method)
- Run regression to get starting values of et and et −1
- Run regression of et = ρet −1
Get ρ̂i
i. Run regression of:
Yt − ρˆ iYt −1 = β 0 (1 − ρˆ i ) + β1 ( X 1t − ρˆ i X 1t )
ii. Run regression of:
et = ρet −1 using new et and et −1
get ρˆ i +1
If ρˆ i +1 − ρˆ i > tolleranceLevel change ρˆ i +1 to ρ̂i and go back to step 1
If ρˆ i +1 − ρˆ i ≤ tolleranceLevel Stop and use your last regression
results
b) The AR(1) method
Estimate a GLS equation like (4) estimating β 0 , β1 , & ρ simultaneously
2. Newey-West standard errors
- Changes standard errors but does not change β j ’s.
o Why does it make sense to adjust se(β j ) but not β j ’s?
- Typically Newey-west standard errors are larger than OLS
standard errors
o What does this do to significance?
Summary:
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