14. Time Series Analysis:

Outline I. II. III. IV. 14. Time Series Analysis: Serial Correlation Read Wooldridge (2013), Chapter 12  V. Properties of OLS with Serially Correla...
Author: Stewart Holmes
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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

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I. Properties II. Testing III. Remedial IV. ARCH V. Hetero & S.C. I. Properties of OLS with Serially Correlated Errors

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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

= 



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Interpret the coefficient on unem.

I. Properties II. Testing III. Remedial IV. ARCH V. Hetero & S.C. II. Testing for Serial Correlation

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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

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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

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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

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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

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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.

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II. Testing for Serial Correlation

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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

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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

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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, 

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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

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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(utxt1, .... 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

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(resid01) to test for serial correlation.

I. Properties II. Testing III. Remedial IV. ARCH V. Hetero & S.C. II. Testing for Serial Correlation

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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.

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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.

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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

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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

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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

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I. Properties II. Testing III. Remedial IV. ARCH V. Hetero & S.C. II. Testing for Serial Correlation

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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

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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

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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 ||