LECTURE 11
Introduction to Econometrics
Autocorrelation
November 29, 2016
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ON PREVIOUS LECTURES
We discussed the specification of a regression equation
Specification consists of choosing:
1. correct independent variables
2. correct functional form
3. correct form of the stochastic error term
We talked about the choice of independent variables and
their functional form
We started to talk about the form of the error term - we
discussed heteroskedasticity
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ON TODAY’S LECTURE
We will finish the discussion of the form of the error term
by talking about autocorrelation (or serial correlation)
We will learn
what is the nature of the problem
what are its consequences
how it is diagnosed
what are the remedies available
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NATURE OF AUTOCORRELATION
Observations of the error term are correlated with each
other
Cov(εi, εj) = 0 , i = j
Violation of one of the classical assumptions
Can exist in any data in which the order of the
observations has some meaning - most frequently in
time-series data
Particular form of autocorrelation - AR(p) process:
εt = ρ1εt−1 + ρ2εt−2 + . . . + ρpεt−p + ut
ut is a classical (not autocorrelated) error term
ρk are autocorrelation coefficients (between -1 and 1)
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EXAMPLES OF PURE AUTOCORRELATION
Distribution of the error term has autocorrelation nature
First order autocorrelation
εt = ρ1εt−1 + ut
positive serial correlation: ρ1 is positive
negative serial correlation: ρ1 is negative
no serial correlation: ρ1 is zero
positive autocorrelation very common in time series data
e.g.: a shock to GDP persists for more than one period
Seasonal autocorrelation (in quarterly data)
εt = ρ4εt−4 + ut
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EXAMPLES OF IMPURE AUTOCORRELATION
Autocorrelation caused by specification error in the
equation:
omitted variable
incorrect functional form
How can misspecification cause autocorrelation in the
error term?
Recall that the error term includes the omitted variables,
nonlinearities, measurement error, and the classical error
term.
If we omit a serially correlated variable, it is included in the
error term, causing the autocorrelation problem.
Impure autocorrelation can be corrected by better choice of
specification (as opposed to pure autocorrelation).
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AUTOCORRELATION
X
Y
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CONSEQUENCES OF AUTOCORRELATION
1. Estimated coefficients (β) remain unbiased and consistent
2. Standard errors of coefficients (s.e.(β)) are biased
(inference is incorrect)
serially correlated error term causes the dependent variable
to fluctuate in a way that the OLS estimation procedure
attributes to the independent variable
Serial correlation typically makes OLS underestimate the
standard errors of coefficients
therefore we find t scores that are incorrectly too high
⇒ The same consequences as for the heteroskedasticity
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DURBIN-WATSON TEST FOR AUTOCORRELATION
Used to determine if there is a first-order serial correlation
by examining the residuals of the equation
Assumptions (criteria for using this test):
The regression includes the intercept
If autocorrelation is present, it is of AR(1) type:
εt = ρεt−1 + ut
The regression does not include a lagged dependent
variable
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DURBIN-WATSON TEST FOR AUTOCORRELATION
Durbin-Watson d statistic (for T observations):
d =
T
t=2
(et − et−1)2
T
t=1
e2
t
≈ 2(1 − ρ)
where ρ is the autocorrelation coefficient
Values:
1. Extreme positive serial correlation: d ≈ 0
2. Extreme negative serial correlation: d ≈ 4
3. No serial correlation: d ≈ 2
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USING THE DURBIN-WATSON TEST
1. Estimate the equation by OLS, save the residuals
2. Calculate the d statistic
3. Determine the sample size T and the number of
explanatory variables (excluding the intercept!) k
4. Find the upper critical value dU and the lower critical
value dL for T and k in statistical tables
5. Evaluate the test as one-sided or two-sided (see next slides)
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ONE-SIDED DURBIN-WATSON TEST
For cases when we consider only positive serial correlation
as an option
Hypothesis:
H0 : ρ ≤ 0 (no positive serial correlation)
HA : ρ > 0 (positive serial correlation)
Decision rule:
if d < dL reject H0
if d > dU do not reject H0
if dL ≤ d ≤ dU inconclusive
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DURBIN-WATSON CRITICAL VALUES FOR ONE-SIDED
TEST
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TWO-SIDED DURBIN-WATSON TEST
For cases when we consider both signs of serial correlation
Hypothesis:
H0 : ρ = 0 (no serial correlation)
HA : ρ = 0 (serial correlation)
Decision rule:
if d < dL reject H0
if d > 4 − dL reject H0
if d > dU do not reject H0
if d < 4 − dU do not reject H0
otherwise inconclusive
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DURBIN-WATSON CRITICAL VALUES FOR TWO-SIDED
TEST
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EXAMPLE
Estimating housing prices in the UK
Quarterly time series data on prices of a representative
house in UK (in £)
Explanatory variable: GDP (in billions of £)
Time span: 1975 Q1 - 2011 Q2
All series are seasonally adjusted and in real prices (i.e.
adjusted for inflation)
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EXAMPLE
60000
80000
100000
120000
140000
160000
180000
200000
220000
1975 1980 1985 1990 1995 2000 2005 2010
Priceofrepresentativehouse
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EXAMPLE
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EXAMPLE
We test for positive serial correlation:
H0 : ρ ≤ 0 (no positive serial correlation)
HA : ρ > 0 (positive serial correlation)
One-sided DW critical values at 95% confidence for
T = 146 and k = 1 are:
dL = 1.72 and dU = 1.74
Decision rule:
if d < 1.72 reject H0
if d > 1.74 do not reject H0
if 1.72 ≤ d ≤ 1.74 inconclusive
Since d = 0.02 < 1.72, we reject the null hypothesis of no
positive serial correlation
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ALTERNATIVE APPROACH TO AUTOCORRELATION
TESTING
Suppose we suspect the stochastic error term to be AR(p)
εt = ρ1εt−1 + ρ2εt−2 + . . . + ρpεt−p + ut
Since OLS is consistent even under autocorrelation, the
residuals are consistent estimates of the stochastic error
term
Hence, it is sufficient to:
1. Estimate the original model by OLS, save the residuals et
2. Regress et = ρ1et−1 + ρ2et−2 + . . . + ρpet−p + ut
3. Test if ρ1 = ρ2 = . . . = ρp = 0 using the standard F-test
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BACK TO EXAMPLE
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BACK TO EXAMPLE
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REMEDY: WHITE ROBUST STANDARD ERRORS
Note that autocorrelation does not lead to inconsistent
estimates, only to incorrect inference - similar to
heteroskedasticity problem
We can keep the estimated coefficients, and only adjust the
standard errors
The White robust standard errors solve not only
heteroskedasticity, but also serial correlation
Note also that all derived results hold if the assumption
Cov(x, ε) = 0 is not violated
First make sure the specification of the model is correct,
only then try to correct for the form of an error term!
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SUMMARY
Autocorrelation does not lead to inconsistent estimates,
but it makes the inference wrong (estimated coefficients
are correct, but their standard errors are not)
It can be diagnosed using
Durbin-Watson test
Analysis of residuals
It can be remedied by
White robust standard errors
Readings:
Studenmund, Chapter 9
Wooldridge, Chapter 12
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