Outline I. II. III. IV.
14. Time Series Analysis: Serial Correlation Read Wooldridge (2013), Chapter 12
V.
Properties of OLS with Serially Correlated Errors Testing for Serial Correlation Correcting for Serial Correlation ARCH (Autoregressive conditional heteroskedasticity) model Heteroskedasticity in Time Series
I. Properties II. Testing III. Remedial IV. ARCH V. Hetero & S.C. 14. Serial Correlation. Quantitative Methods of Economic Analysis . 2949605 . Chairat Aemkulwat
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14. Serial Correlation. Quantitative Methods of Economic Analysis . 2949605 . Chairat Aemkulwat
Serial Correlation
I. Properties of OLS with Serially Correlated Errors
Positive Serial Correlation
• If there is the problem of serial correlation, i.e., for s t E(utus|xt, xs) 0 E(utut‐1) 0 for AR(1) • Serial Correlation means that errors are correlated. (See Graphs)
Negative Serial Correlation
• Static and finite distributed lag models often have serially correlated errors.
I. Properties II. Testing III. Remedial IV. ARCH V. Hetero & S.C. I. Properties of OLS with Serially Correlated Errors
14. Serial Correlation. Quantitative Methods of Economic Analysis . 2949605 . Chairat Aemkulwat
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I. Properties II. Testing III. Remedial IV. ARCH V. Hetero & S.C. I. Properties of OLS with Serially Correlated Errors
14. Serial Correlation. Quantitative Methods of Economic Analysis . 2949605 . Chairat Aemkulwat
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Unbiasedness and Serial Correlation
Errors follow the AR(1) process
• Unbiasedness (TS.1 ‐ TS.3) In the presence of serial correlation, OLS estimators are unbiased in finite samples.
• Consider the model yt = 0 + 1xt + ut
• Consistency (TS.1‐TS.3) In the presence of serial correlation, OLS estimators are consistent in large samples.
ut = ut‐1 + et. where (i) {ut} follow the AR(1) process and (ii) et ~ iid(0, e2) and|| 2, then H0: =0 is rejected. We conclude that there is first‐order serial correlation.
I. Properties II. Testing III. Remedial IV. ARCH V. Hetero & S.C.
II. Testing for Serial Correlation
t ‐1
to find the (coefficient of statistic.
• H0: =0 or the null hypothesis is that there is no serial correlation. If the null is true, then TS.5 is true.
II. Testing for Serial Correlation
=
•
11
Interpret the coefficient on unem.
I. Properties II. Testing III. Remedial IV. ARCH V. Hetero & S.C. II. Testing for Serial Correlation
14. Serial Correlation. Quantitative Methods of Economic Analysis . 2949605 . Chairat Aemkulwat
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Step 1: Regress inf on unem
Static Phillips
Step 2: Eviews: Proc/Make Residual Series (named resid01) Dependent Variable: INF
Step 2 : Use (residuals) to test for serial correlation. Eviews: Proc/Make Residuals Series (resid01)
Method: Least Squares Sample: 1 49 Included observations: 49
Step 3: Test for serial correlation. Regress t on t‐1 to find and t value. 01 = .573resid01(‐1) (s.e) (.115013) {t} {4.9797}
Std. Error
t-Statistic
Prob.
1.42361
1.719015
0.828154
0.4118
UNEM
0.467626
0.289126
1.617376
0.1125
0.052723
13
14. Serial Correlation. Quantitative Methods of Economic Analysis . 2949605 . Chairat Aemkulwat
Step 3: Obtain (resid01) to test for serial correlation by regressing resid01 on resid01(‐1)
Mean dependent var
4.108163
Adjusted R-squared
0.032568
S.D. dependent var
3.182821
S.E. of regression
3.130562
Akaike info criterion
5.160262
Sum squared resid
460.6198
Schwarz criterion
5.237479
Log likelihood
-124.426
F-statistic
2.615904
Durbin-Watson stat
I. Properties II. Testing III. Remedial IV. ARCH V. Hetero & S.C.
0.8027
Prob(F-statistic)
0.11249
I. Properties II. Testing III. Remedial IV. ARCH V. Hetero & S.C. II. Testing for Serial Correlation
14. Serial Correlation. Quantitative Methods of Economic Analysis . 2949605 . Chairat Aemkulwat
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Example: Expectations Augmented Phillips Curve • inft – infte = 1(unemt ‐ 0) + et
Dependent Variable: RESID01 Method: Least Squares
0
Sample(adjusted): 2 49
infte unemt ‐ 0 inft – infte et
Included observations: 48 after adjusting endpoints Variable
Coefficient
Std. Error
t-Statistic
Prob.
RESID01(-1)
0.572735
0.115013
4.979738
0
R-squared
0.344633
Mean dependent var
-0.10207
Adjusted R-squared
0.344633
S.D. dependent var
3.046154
S.E. of regression
2.466005
Akaike info criterion
Sum squared resid
285.8156
Schwarz criterion
4.702673
Log likelihood
-110.929
Durbin-Watson stat
1.351045
4.66369
I. Properties II. Testing III. Remedial IV. ARCH V. Hetero & S.C. II. Testing for Serial Correlation
Coefficient
C R-squared
• Is there any problem of serial correlation?
II. Testing for Serial Correlation
Variable
14. Serial Correlation. Quantitative Methods of Economic Analysis . 2949605 . Chairat Aemkulwat
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: natural rate of unemployment : expected rate of inflation : cyclical unemployment : unanticipated inflation : supply shock (error term)
• Adaptive Expectation theory says that the expected value of current inflation depends on recently observed inflation. Let infte = inft‐1 inft = 0 + 1unemt + et I. Properties II. Testing III. Remedial IV. ARCH V. Hetero & S.C. II. Testing for Serial Correlation
14. Serial Correlation. Quantitative Methods of Economic Analysis . 2949605 . Chairat Aemkulwat
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Results: Expectations Augmented Phillips Curve
Results: Expectations Augmented Phillips Curve
Step 1: regress inf‐inf(‐1) on unem
Step 3: Test for serial correlation. Regress t on t‐1 to find and t‐stat
t = 3.03 ‐ 0.543umemt (s.e) (1.38) (0.280) [t‐stat] [2.2] [‐2.35] n = 48 R2 = 0.108 R2 bar= 0.088
02 = ‐.0357resid02(‐1) (s.e) {.123} [t‐stat] [‐.289]
*Interpret the coefficient of unem.
• Is there any problem of serial correlation?
Step 2: Use (residuals) to test for serial correlation – resid012
I. Properties II. Testing III. Remedial IV. ARCH V. Hetero & S.C. II. Testing for Serial Correlation
14. Serial Correlation. Quantitative Methods of Economic Analysis . 2949605 . Chairat Aemkulwat
I. Properties II. Testing III. Remedial IV. ARCH V. Hetero & S.C.
17
II. Testing for Serial Correlation
14. Serial Correlation. Quantitative Methods of Economic Analysis . 2949605 . Chairat Aemkulwat
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Step 1: Use D(inf) in place of inf‐inf(‐1) in Eviews
Step 1: Estimation output: Regress inf‐inf(‐1) on unem
Step 2: Eviews: Proc/Make Residual Series (name resid02)
Step 2: Eviews: Proc/Make Residual Series (name resid02)
Dependent Variable: D(INF) Dependent Variable: INF-INF(-1)
Method: Least Squares
Sample(adjusted): 2 49
Sample(adjusted): 2 49
Included observations: 48 after adjusting endpoints
Included observations: 48 after adjusting endpoints
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
3.030581
1.37681
2.201161
0.0328
C
UNEM
-0.54259
0.230156
-2.357475
0.0227
UNEM
R-squared Adjusted R-squared
0.107796 0.0884
Variable
Coefficient 3.030581
Mean dependent var
-0.10625
R-squared
S.D. dependent var
2.566926
Adjusted R-squared
-0.542587 0.107796 0.0884
Std. Error
t-Statistic
1.37681
2.201161
0.230156
-2.357475
Prob. 0.0328 0.0227
Mean dependent var
-0.10625
S.D. dependent var
2.566926
Akaike info criterion
4.671515
S.E. of regression
2.450843
Akaike info criterion
4.671515
S.E. of regression
Sum squared resid
276.3051
Schwarz criterion
4.749482
Sum squared resid
276.3051
Schwarz criterion
4.749482
Log likelihood
-110.116
F-statistic
5.557689
Log likelihood
-110.1164
F-statistic
5.557689
Durbin-Watson stat
1.769648
Prob(F-statistic)
0.02271
I. Properties II. Testing III. Remedial IV. ARCH V. Hetero & S.C. II. Testing for Serial Correlation
14. Serial Correlation. Quantitative Methods of Economic Analysis . 2949605 . Chairat Aemkulwat
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Durbin-Watson stat
2.450843
1.769648
Prob(F-statistic)
0.02271
I. Properties II. Testing III. Remedial IV. ARCH V. Hetero & S.C. II. Testing for Serial Correlation
14. Serial Correlation. Quantitative Methods of Economic Analysis . 2949605 . Chairat Aemkulwat
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Step 2: Test for Serial Correlation. Regress
t
on
Durbin‐Watson Test
t‐1
• The Durbin‐Watson Test is valid under classical assumptions only. (TS.1‐TS.6 except TS.5) • Durbin Watson (DW) Statistic is
Dependent Variable: RESID02 Method: Least Squares Sample(adjusted): 3 49
n
Included observations: 47 after adjusting endpoints Variable
Coefficient
RESID02(-1)
-0.03566
R-squared
-0.00744
Std. Error 0.123104
t-Statistic -0.289695
DW
Prob. 0.194241
-0.00744
S.D. dependent var
2.038688
S.E. of regression
2.046255
Akaike info criterion
4.290946
Sum squared resid
192.6093
Schwarz criterion
4.330311
Log likelihood
-99.8372
Durbin-Watson stat
1.827911
I. Properties II. Testing III. Remedial IV. ARCH V. Hetero & S.C. 14. Serial Correlation. Quantitative Methods of Economic Analysis . 2949605 . Chairat Aemkulwat
t 1
uˆ t
2
t‐1
2 and with moderate sample sizes,
14. Serial Correlation. Quantitative Methods of Economic Analysis . 2949605 . Chairat Aemkulwat
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Examples •
Example: Static Phillips Model = 1.423+ .468unem n=49 R2=.053 R2bar=.032 Durbin‐Watson stat = 0.80 Table (n=50, k=1, =1%); dL=1.324; dU=1.403 DW2, then we reject H0. I. Properties II. Testing III. Remedial IV. ARCH V. Hetero & S.C. II. Testing for Serial Correlation
26
14. Serial Correlation. Quantitative Methods of Economic Analysis . 2949605 . Chairat Aemkulwat
Example: Static Phillips Revisited • inft = 0 + 1unemt + ut
LM = (n‐q)R2u^ ~ 2q
• Step 1: Estimation output
where q is the order number of an autoregressive process [AR(q)] in the null hypothesis. – What are the rejection rule? – What is the number of restrictions?
= 1.423+ .468unem (s.e) (1.72) (.289) [t] [.828] {1.617} n=49 R2=.053 R2bar=.032
• The test requires the homoskedasticity assumption Var(utxt1, .... xtk, t‐1) = 2
14. Serial Correlation. Quantitative Methods of Economic Analysis . 2949605 . Chairat Aemkulwat
,
and obtain , the coefficient of
• Breusch‐Godfrey LM Test
I. Properties II. Testing III. Remedial IV. ARCH V. Hetero & S.C.
Durbin suggested to run the regression of t on xt1, .... xtk
t‐test and LM Test
II. Testing for Serial Correlation
Durbin (1970). Given a model,
Step 2: Use
27
(resid01) to test for serial correlation.
I. Properties II. Testing III. Remedial IV. ARCH V. Hetero & S.C. II. Testing for Serial Correlation
14. Serial Correlation. Quantitative Methods of Economic Analysis . 2949605 . Chairat Aemkulwat
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Step 1: Regress inf on unem
Example: Static Phillips Revisited • Step 3: Regress
Step 2: Eviews: Proc/Make Residual Series (named resid01)
on unem and t‐1 01 = 2.157 ‐.3929unem + .6449resid01(‐1) {t} {5.247} 2 n=48, R =.380747
Dependent Variable: INF
t
Method: Least Squares Sample: 1 49 Included observations: 49 Variable
• Interpretation: 1) Do we reject the null hyptothesis according to t Test? 2) LM Test: LM = (n‐q)*R‐squared = (49‐1)*.380747 = 18.29 c= 6.63 is the 99th percentile in the distribution of 21.
I. Properties II. Testing III. Remedial IV. ARCH V. Hetero & S.C.
Std. Error
t-Statistic
Prob.
C
1.42361
1.719015
0.828154
0.4118
UNEM
0.467626
0.289126
1.617376
0.1125
R-squared
0.052723
Mean dependent var
4.108163
Adjusted R-squared
0.032568
S.D. dependent var
3.182821
S.E. of regression
3.130562
Akaike info criterion
5.160262
Sum squared resid
460.6198
Schwarz criterion
5.237479
Log likelihood
-124.426
F-statistic
2.615904
Durbin-Watson stat
II. Testing for Serial Correlation
Coefficient
0.8027
Prob(F-statistic)
0.11249
I. Properties II. Testing III. Remedial IV. ARCH V. Hetero & S.C.
29
II. Testing for Serial Correlation
14. Serial Correlation. Quantitative Methods of Economic Analysis . 2949605 . Chairat Aemkulwat
Step 3: Regress t on unem and t‐1 to find in the case that unem is not strictly exogenous.
14. Serial Correlation. Quantitative Methods of Economic Analysis . 2949605 . Chairat Aemkulwat
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Eviews : the case of nonstrictly exogeneous regressors In the “equation” window (step 1), choose View/Residual Tests/ Serial Correlation LM Test. In the “lag Specification” box, type in “1” for one lag in residuals.
Dependent Variable: RESID01 Method: Least Squares Sample(adjusted): 2 49
Breusch-Godfrey Serial Correlation LM Test:
Included observations: 48 after adjusting endpoints Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
2.157068
1.472917
1.464487
0.15
UNEM
-0.392975
0.247479
-1.587912
0.1193
RESID01(-1)
0.644904
0.122912
5.246876
0 -0.10207
Adjusted R-squared
0.353224
S.D. dependent var
3.046154
S.E. of regression
2.449789
Akaike info criterion
4.690342
Sum squared resid
270.0659
Schwarz criterion
4.807292
F-statistic
13.83408
1.358907
Prob(F-statistic)
0.000021
I. Properties II. Testing III. Remedial IV. ARCH V. Hetero & S.C. II. Testing for Serial Correlation
14. Serial Correlation. Quantitative Methods of Economic Analysis . 2949605 . Chairat Aemkulwat
0.000003 0.000017
Presample missing value lagged residuals set to zero.
Mean dependent var
Durbin-Watson stat
Probability Probability
Dependent Variable: RESID
0.380747
-109.5682
27.83291 18.47161
Test Equation:
R-squared
Log likelihood
F-statistic Obs*R-squared
31
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
2.705871
1.464288
1.847909
0.0711
UNEM
-0.47356
0.24753
-1.913156
0.062
RESID(-1)
0.659484
0.125004
5.27569
0
R-squared
0.376972
Mean dependent var
-1.97E-16
Durbin-Watson stat
1.818217
Prob(F-statistic)
0.000019
I. Properties II. Testing III. Remedial IV. ARCH V. Hetero & S.C. II. Testing for Serial Correlation
14. Serial Correlation. Quantitative Methods of Economic Analysis . 2949605 . Chairat Aemkulwat
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Example: Expectations Augmented Phillips Curve Revisited
Testing for Higher Order Serial Correlation •
Test serial correlation in the autoregressive of order q or AR(q) ut = 1ut‐1 + 2ut‐2 + ... +qut‐q + et
•
Test the null hypothesis H0: 1= 2= ... = q= 0
•
We can run the regression of t on xt1, .... , xtk , ut‐1 , .... , ut‐q
•
• Regress D(inf) on unem
t = 3.03 ‐ 0.543umemt n = 48 R2 = 0.108 R2 bar= 0.088
• Suppose we wish to test H0: 1= 2= 0 in the AR(2) model ut = 1ut‐1 + 2ut‐2+ unem + et
Compute the F test for joint significance of ut‐1 , .... , ut‐q or the LM statistic is LM = (n‐q)R2u^ ~ 2q
• Run the regression of resid c unem resid(‐1) resid(‐2)
This is called the Breusch‐Godfrey test for AR(q) serial correlation I. Properties II. Testing III. Remedial IV. ARCH V. Hetero & S.C. II. Testing for Serial Correlation
14. Serial Correlation. Quantitative Methods of Economic Analysis . 2949605 . Chairat Aemkulwat
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I. Properties II. Testing III. Remedial IV. ARCH V. Hetero & S.C. II. Testing for Serial Correlation
14. Serial Correlation. Quantitative Methods of Economic Analysis . 2949605 . Chairat Aemkulwat
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In the “equation” window (step 1) that regress inf on unem, choose View/Residual Tests/ Serial Correlation LM Test. In the “lag Specification” box, type in “2” for two lags in residuals.
Eviews: Expectations Augmented Phillips Curve Revisited
Breusch-Godfrey Serial Correlation LM Test:
• In the “equation” window (step 1), choose View/Residual Tests/ Serial Correlation LM Test. In the “lag Specification” box, type in “2” for one lag in residuals.
F-statistic
4.408984
Probability
0.017979
Obs*R-squared
8.013607
Probability
0.018191
Test Equation: Dependent Variable: RESID Method: Least Squares Presample missing value lagged residuals set to zero.
• H0: 1= 2= 0 F‐statistic 4.409 (p‐value=.017979) LM‐statisic 8.0136 (p‐value=.0018191) We reject the null hypothesis, so there is strong evidence of AR(2) serial correlation. Compare to AR(1)!
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
-0.83592
1.31912
-0.633697
0.5296
UNEM
0.144109
0.220873
0.652453
0.5175
RESID(-1)
-0.05971
0.138511
-0.431046
0.6685
RESID(-2)
-0.4168
0.140903
-2.958062
0.005
R-squared
I. Properties II. Testing III. Remedial IV. ARCH V. Hetero & S.C. II. Testing for Serial Correlation
14. Serial Correlation. Quantitative Methods of Economic Analysis . 2949605 . Chairat Aemkulwat
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0.16695
Mean dependent var
4.39E-16
I. Properties II. Testing III. Remedial IV. ARCH V. Hetero & S.C. II. Testing for Serial Correlation
14. Serial Correlation. Quantitative Methods of Economic Analysis . 2949605 . Chairat Aemkulwat
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1.1) Find best linear unbiased estimators (BLUE) in the AR(1) model
III. Correcting for serial correlation Remedial measures: After detecting serial correlation, we now learn how to fix the problem.
• Assume the errors {ut} follow the AR(1) model:
1) Assume strictly exogenous regressors 1.1 Known : find generalized least squares (GLS) estimators 1.2 Use estimated ‐ Feasible GLS (FGLS) • Cochrance‐Orcutt Method • Iterated Cochrance‐Orcutt Method • Differencing
ut = ut‐1 + et where et ~ i.i.d(0, e2) and ||