chapter 9 correlated errors...auxiliary regression of residuals on lag test statistic: critical...
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Chapter 9Correlated Errors
Learning Objectives• Demonstrate the problem of correlated errors and its
implications
• Conduct and interpret tests for correlated errors
• Correct for correlated errors using Newey and West’s estimator (ex post) or using generalized least squares (ex ante)
• Correct for correlated errors by adding lagged variables to the model
• Show that correlated errors can arise in clustered and spatial data as well as in time-series data
Autocorrelated Errors
-20
-10
0
10
20
30
40
50
1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015
Coca Cola Stock Price
Logs Make the Model Fit Better
-5
-4
-3
-2
-1
0
1
2
3
4
5
1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015
Log of Coca Cola Stock Price
Autocorrelated Errors
-1.5
-1
-0.5
0
0.5
1
1.5
2
1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015
Log of Coca Cola Stock Price: Deviations from Trend
The Problem in Words• You think you have more information in your data
than you really do.• It can be the opposite – you think you have less info than you really do, but this is
rare
• OLS estimates unbiased, but not BLUE
• Causes standard errors to be underestimated
• Examples1. Stock prices over time2. Consumption over time3. Income across space
The Problem Mathematically
• If ρ > 0, then some of last period’s error remains in this period’s error – we have less new information each period than the standard error formula assumes.
• If ρ < 0, then we have more information than the s.e. formula assumes – this is rare!
0 1 1 2 2
1
β β β εε ρε −
= + + += +
t t t t
t t t
Y X Xu
Solutions
1. Test and fix after-the-fact (ex-post)
2. Change the model to eliminate the correlated errors
i. Generalized least squares (ex-ante correction)ii. Change the model by adding lagged variables (best
approach)
Testing for AutocorrelationAutocorrelation Model
Breusch-Godfrey test1. Estimate regression
2. Auxiliary regression of residuals on lag
0 1 1 2 2
1
β β β εε ρε −
= + + += +
t t t t
t t t
Y X Xu
1 0 1 1 2 2ρ α α α−= + + + +t t t t te e X X u
0 1 1 2 2= + + +t t t tY b b X b X e
Breusch-Godfrey TestAuxiliary regression of residuals on lag
Test statistic:Critical value: (e.g., at 5% significance, c.v. = 3.84)
Can add more lags to auxiliary regression
critical value:
1 0 1 1 2 2ρ α α α−= + + + +t t t t te e X X u
2( 1)= −BG T R2(1)χ
1 1 2 2 0 1 1 2 2...ρ ρ ρ α α α− − −= + + + + + + +t t t m t m t t te e e e X X u2( )= −BG T m R 2
( )χ m
Example: US Consumption vs Income
0
2000
4000
6000
8000
10000
12000
1947 1957 1967 1977 1987 1997 2007
$ pe
r qu
arte
r
YearReal personal consumption expenditures per capitaReal disposable personal income per capita
Taking Logs Straightens the Trend
7
7.5
8
8.5
9
9.5
1947 1957 1967 1977 1987 1997 2007
natu
ral l
og o
f $ p
er q
uart
er
Year
Log real personal consumption expenditures per capitaLog real disposable personal income per capita
Regression Residuals
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
1947 1957 1967 1977 1987 1997 2007
Year
ln( ) 0.38 1.03ln( )= + −t t te cons income
There appears to be strong autocorrelation
Breusch-Godfrey Test
VariableEstimated Coefficient
Standard Error
t-statistic
e(t-1) 0.885 0.027 32.70Ln(Income) 0.001 0.001 0.56
Constant -0.007 0.012 -0.58
Sample Size 274
R-squared 0.80
(T-1)*R-squared 217.80
Critical χ2(1) 1 df, 5% 3.84
We reject the null hypothesis at the 5% level – we have autocorrelation
Newey-West Correction for Standard Errors
If CR2 and CR3 Hold:
If CR2 fails (White’s Method)
If CR2 and CR3 fail (Newey-West)
2 21
1[ ]
=
=∑N
ii
V b w s
2 21
1[ ]
=
=∑N
i ii
V b w e
2 21
1 1[ ] 1 2 1
= =
= + −
∑ ∑T L
t t jt j
jV b w e rL
Changing the Model: GLSAutocorrelation Model
The error ut satisfies CR2 and CR3
We have a new model
0 1
1
β β εε ρε −
= + += +
t t t
t t t
Y Xu
( )0 1 1
0 1 1 0 1 1
β β ρεβ β ρ β β
−
− −
= + + +
= + + − − +t t t t
t t t t
Y X uX Y X u
* * *0 1β β= + +t t tY X u
( )1 0 1 1(1 )ρ β ρ β ρ− −− = − + − +t t t t tY Y X X u
Changing the Model: GLSWe have a new model
where and
• But we don’t know ρ. Solution: “Feasible GLS”
1. OLS regression
2. Error autocorrelation
3. Transform variables
4. OLS regression
* * *0 1β β= + +t t tY X u
*1ρ −= −t t tY Y Y *
1ρ −= −t t tX X X
0 1= + +t t tY b b X e2
1 12 2
− −= =
=∑ ∑T T
t t tt t
r e e e
* *1 1,− −= − = −t t t t t tY Y rY X X rX
* *0 1= + +GLS GLS GLS
t t tY b b X e
OLS with Newey-West vs FGLS
Variable
OLS (Newey West std. error with L=40)
FGLS
Estimated Coefficient
Standard Error
Estimated Coefficient
Standard Error
Income 1.030 0.015 1.012 0.013Constant -0.379 0.132 -0.026 0.012
Sample Size 275 274r 0.88
Changing the Model: Distributed LagsAutocorrelation Model
The error ut satisfies CR2 and CR3
We have a new model
0 1
1
β β εε ρε −
= + += +
t t t
t t t
Y Xu
( )0 1 1
0 1 1 0 1 1
0 1 1 1 1(1 )
β β ρεβ β ρ β ββ ρ ρ β ρβ
−
− −
− −
= + + +
= + + − − +
= − + + − +
t t t t
t t t t
t t t t
Y X uX Y X u
Y X X u
* * * *0 1 1 2 3 1β β β β− −= + + + +t t t t tY Y X X u
Distributed Lag Model
Interpreting the Model: Two Tricks
1. Drop the time subscripts to get “long-run”
* * * *0 1 1 2 3 1β β β β− −= + + + +t t t t tY Y X X u
* * * *0 1 2 3
* * *0 2 3
1 11 * 1 *
β β β β
β β ββ β
= + + +
+= +
− −
Y Y X X
Y X
Distributed Lag Model
Interpreting the Model: Two Tricks1. Drop the time subscripts to get “long-run”
2. Derive the “error correction model”
* * * *0 1 1 2 3 1β β β β− −= + + + +t t t t tY Y X X u
( )
* * * * * *1 0 1 1 2 3 1 1 3 3
* * ** *0 2 31 1 3 1* *
1 1
(1 )1 1
β β β β β β
β β ββ ββ β
− − − −
− −
− = + + + + − + −
+= − − − − − − + − −
t t t t t t t t t
t t t t t
Y Y Y X X u Y X X
Y X X X u
* * *0 2 3
1 11 * 1 *β β ββ β
+= +
− −Y X
Distributed Lag Estimates
VariableEstimated Coefficient
Standard Error
t-statistic
Consumption(t-1) 0.927 0.020 47.20Income 0.297 0.046 6.44
Income(t-1) -0.222 0.048 -4.66Constant -0.021 0.012 -1.75
Sample Size 274R-squared 0.9997
Distributed Lag Model
Interpreting the Model: Two Tricks
1. Drop the time subscripts to get “long-run”
2. Derive the “error correction model”( ) ( )1 1 10.073 0.29 1.03 0.222− − −− = − + − + − +t t t t t t tY Y Y X X X u
0.29 1.03= − +Y X
1 10.021 0.927 0.297 0.222− −= − + + − +t t t t tY Y X X u
Correlated Errors Across Space• Famous study by Brent Moulton in 1990• Data: (i) Wages for 18,946 workers in the US
(ii) 14 garbage variables• Moulton regressed wages on the 14 garbage
variables plus education and work experience– 6 of the 14 garbage variables were significant
• WHY? Spatial correlation in errors made standard errors 3-5 times too small
“Clustering” Standard Errors
Correlation over time: Newey-West
Correlation over space: Clustering
2 21
1 1[ ] 1 2 1
= =
= + −
∑ ∑T L
t t jt j
jV b w e rL
11 1
[ ] 1( , in same cluster)= =
=∑∑N N
i i j ji j
V b i j w e w e
What We Learned
• Correlated errors cause OLS to lose its “best” property and the estimated standard errors to be biased.
• Same as heteroskedasticity
• As long as the autocorrelation is not too strong, the standard error bias can be corrected with Newey and West’s heteroskedasticity and autocorrelation consistent estimator.
• Getting the model right by adding lagged variables to the model is usually the best approach to deal with autocorrelation in time-series data.
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