14. time series analysis: i. ols serial correlation ii. iii. iv....

14. Time Series Analysis:Serial Correlation

Read Wooldridge (2013), Chapter 12

14. Serial Correlation. Quantitative Methods of Economic Analysis . 2949605 . Chairat Aemkulwat

Outline

I. Properties of OLS with Serially Correlated ErrorsII. Testing for Serial CorrelationIII. Correcting for Serial CorrelationIV. ARCH (Autoregressive conditional

heteroskedasticity) modelV. Heteroskedasticity in Time Series

2I. Properties II. Testing III. Remedial IV. ARCH V. Hetero & S.C.14. Serial Correlation. Quantitative Methods of Economic Analysis . 2949605 . Chairat Aemkulwat

I. Properties of OLS with Serially Correlated Errors

• If there is the problem of serial correlation, i.e.,E(utus|xt, xs) 0 for s tE(utut‐1) 0 for AR(1)

• Serial Correlation means that errors are correlated. (See Graphs)

• Static and finite distributed lag models often have serially correlated errors.

3I. Properties II. Testing III. Remedial IV. ARCH V. Hetero & S.C.14. Serial Correlation. Quantitative Methods of Economic Analysis . 2949605 . Chairat Aemkulwat I. Properties of OLS with Serially Correlated Errors

Positive Serial

Correlation

Negative Serial

Correlation

Serial Correlation


Unbiasedness and Serial Correlation

• Unbiasedness (TS.1 ‐ TS.3)In the presence of serial correlation, OLS estimators are unbiased in finite samples.

• Consistency (TS.1‐TS.3)In the presence of serial correlation, OLS estimators are consistent in large samples.

• If there is the problem of serial correlation, the Gauss Markov Theorem no longer holds.– Note that Gauss Markov Theorem (TS.1‐TS.5) requires no

heteroskedasticity (TS.4) and no serial correlation (TS.5).


Errors follow the AR(1) process

• Consider the model

yt = 0 + 1xt + ut

ut = ut‐1 + et.

where (i) {ut} follow the AR(1) process and(ii) et ~ iid(0, e

2) and|| < 1).

• Assume TS.1‐TS.4 hold1) and are unbiased and consistent.2) OLS estimators are no longer BLUE.


Errors follow the AR(1) process

Assume TS.1‐TS.4 hold3) Variance of is

4) s.e.( ) is invalid.– Usually, OLS s.e. will underestimate the true s.e. when

estimating using Eviews.– Thus, the usual t statistic is invalid. Moreover, F and LM

statistics are also invalid.

jtt

tn

j

jn

txx

xxSSTSST

Var

1

1

12

22

12)ˆ(


II. Testing for Serial Correlation

Methods1. When regressors are strictly exogenous.

• t‐test for AR(1)• Durbin‐Watson Statistic

2. When regressors are not strictly exogenous.• t‐test for AR(1)• F‐test for AR(q)• Breusch‐Godfrey LM serial correlation Test

8I. Properties II. Testing III. Remedial IV. ARCH V. Hetero & S.C.

Strict exogeneity implies that ut is uncorrelated with regressors for all time periods.

14. Serial Correlation. Quantitative Methods of Economic Analysis . 2949605 . Chairat Aemkulwat II. Testing for Serial Correlation

1. Test for AR(1) serial correlation with strictly exogenous regressors


yt = 0 + 1xt1 + 2xt2 + … + kxtk + utut = ut‐1 + et [ut are weakly dependent]

where (i) {ut} follow the AR(1) process and(ii) {et} ~ iid(0, e

2) and|| < 1

• H0: =0 or the null hypothesis is that there is no serial correlation. If the null is true, then TS.5 is true.

9I. Properties II. Testing III. Remedial IV. ARCH V. Hetero & S.C.14. Serial Correlation. Quantitative Methods of Economic Analysis . 2949605 . Chairat Aemkulwat II. Testing for Serial Correlation

Use residuals for errors• Given AR(1) process

ut = ut‐1 + et

• Estimation:

t = t ‐1 + et

to find the (coefficient of t ‐1) and its corresponding t statistic.

• Rejection rule: If t >2, then H0: =0 is rejected. We conclude that there is first‐order serial correlation.


Notes on t‐test for AR(1) serial correlation

1) This is a large sample test. is the consistent estimator of .

(estimation error in )

2) Practical and statistical significance.If t>2 but is small, then OLS inference procedures are not far off.

3) This t test detects serial correlation of adjacent terms, AR(1) in ut


Example: Static Phillips Model

• Phillips curve shows the tradeoff between inflation and unemployment.

inft = 0 + 1unemt + ut

• Step 1: Estimation output= 1.423+ .468unem

(s.e) (1.72) (.289)[t] [.828] {1.617}

n=49 R2=.053 R2bar=.032

• Interpret the coefficient on unem.


Static Phillips

Step 2 : Use (residuals) to test for serial correlation.Eviews: Proc/Make Residuals Series (resid01)

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}

• Is there any problem of serial correlation?


Dependent Variable: INF

Method: Least Squares

Sample: 1 49

Included observations: 49

Variable Coefficient 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 0.8027 Prob(F-statistic) 0.11249

Step 1: Regress inf on unem

Step 2: Eviews: Proc/Make Residual Series (named resid01)


Dependent Variable: RESID01Method: Least SquaresSample(adjusted): 2 49Included 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 -0.10207Adjusted R-squared 0.344633 S.D. dependent var 3.046154S.E. of regression 2.466005 Akaike info criterion 4.66369Sum squared resid 285.8156 Schwarz criterion 4.702673Log likelihood -110.929 Durbin-Watson stat 1.351045

Step 3: Obtain (resid01) to test for serial correlation by regressing resid01 on resid01(‐1)


Example: Expectations Augmented Phillips Curve

• inft – infte = 1(unemt ‐ 0) + et

0 : natural rate of unemploymentinfte : expected rate of inflationunemt ‐ 0 : cyclical unemploymentinft – infte : unanticipated inflationet : supply shock (error term)

• Adaptive Expectation theory says that the expected value of current inflation depends on recently observed inflation.

Let infte = inft‐1inft = 0 + 1unemt + et


Step 1: regress inf‐inf(‐1) on unem

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

*Interpret the coefficient of unem.

Step 2: Use (residuals) to test for serial correlation – resid012


Results: Expectations Augmented Phillips Curve


Results: Expectations Augmented Phillips Curve

Step 3: Test for serial correlation.Regress t on t‐1 to find and t‐stat

02 = ‐.0357resid02(‐1)(s.e) {.123}[t‐stat] [‐.289]

• Is there any problem of serial correlation?


Dependent Variable: INF-INF(-1)Sample(adjusted): 2 49

Included observations: 48 after adjusting endpoints

Variable Coefficient Std. Error t-Statistic Prob. C 3.030581 1.37681 2.201161 0.0328

UNEM -0.54259 0.230156 -2.357475 0.0227R-squared 0.107796 Mean dependent var -0.10625






Step 1: Estimation output: Regress inf‐inf(‐1) on unem

Step 2: Eviews: Proc/Make Residual Series (name resid02)


Dependent Variable: D(INF)


Sample(adjusted): 2 49



C 3.030581 1.37681 2.201161 0.0328

UNEM -0.542587 0.230156 -2.357475 0.0227

R-squared 0.107796 Mean dependent var -0.10625






Step 1: Use D(inf) in place of inf‐inf(‐1) in Eviews


Step 2: Eviews: Proc/Make Residual Series (name resid02)



Variable Coefficient Std. Error t-Statistic Prob. RESID02(-1) -0.03566 0.123104 -0.289695 0.7734

R-squared -0.00744 Mean dependent var 0.194241Adjusted R-squared -0.00744 S.D. dependent var 2.038688S.E. of regression 2.046255 Akaike info criterion 4.290946Sum squared resid 192.6093 Schwarz criterion 4.330311Log likelihood -99.8372 Durbin-Watson stat 1.827911

Step 2: Test for Serial Correlation. Regress t on t‐1


Durbin‐Watson Test

• The Durbin‐Watson Test is valid under classical assumptions only. (TS.1‐TS.6 except TS.5)

• Durbin Watson (DW) Statistic is

• Simple Algebra shows that DW and are closely linked

DW 2(1‐ ) (*)

using the fact that t2 t‐1

2 and with moderate sample sizes, equation (*) is pretty close.

n

1t

2t

n

2t1tt

u

uuˆ

n

1t

2t

n

2t

21tt

u

)uu(DW


Rejection Rule:• Null Hypothesis: H0: =0. Durbin‐Watson table gives

dL: lower bounddu: upper bound

• Choose significance level (=.05) and need to know

n: sample size k # of regressors

• Rejection Rule: the case of positive serial correlation

if DW<dL, H0 is rejected in favor of H1: >0if DW>dU, we do not reject H0 dL<DW<du, we don’t know (inconclusive)


• Example: Static Phillips Model

= 1.423+ .468unemn=49 R2=.053 R2bar=.032

Durbin‐Watson stat = 0.80Table (n=50, k=1, =1%); dL=1.324; dU=1.403DW< dL or .80<1.324 ; there is an evidence of serial correlation

• Example: Expectations Augmented Phillips Model

t = 3.03 ‐ 0.543umemtn = 48, R2 = 0.108, R2 bar= 0.088

Durbin‐Watson stat = 1.80Table (n=45, k=1, =5%); dL=1.48; dU=1.57DW > dUor 1.80>1.57 ; there is no evidence of serial correlation.


Examples


• Advantage of DW Test: exact sampling distribution for DW can be tabulated.

• Disadvantages of DW Test: DW is valid under the full set of CLM assumptions.

Advantages of t Test: • t statistic is asymptotically valid even without TS.6

assumption.• t statistic is valid in the presence of heteroskedasticity that

depends on the xij.


Advantages and Disadvantages


2. Testing for AR(1) Serial Correlation without Assuming Strict Exogeneity

• Durbin (1970). Given a model,

yt = 0 + 1xt1+ 2xt2+ .... + kxtk + ut

where xij could contain lagged dependent variables (eg. yt‐1).

• Durbin suggested to run the regression of

ton xt1, .... xtk , t‐1

and obtain , the coefficient of t‐1 and its corresponding t value.

• Null Hypothesis: H0: =0 t‐test: Rejection Rule: If t>2, then we reject H0.


t‐test and LM Test

• Breusch‐Godfrey LM Test

LM = (n‐q)R2u^ ~ 2q

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?

• The test requires the homoskedasticity assumptionVar(utxt1, .... xtk, t‐1) = 2


• inft = 0 + 1unemt + ut

• Step 1: Estimation output

= 1.423+ .468unem(s.e) (1.72) (.289)[t] [.828] {1.617}

n=49 R2=.053 R2bar=.032

Step 2: Use (resid01) to test for serial correlation.

28

Example: Static Phillips Revisited

I. Properties II. Testing III. Remedial IV. ARCH V. Hetero & S.C.14. Serial Correlation. Quantitative Methods of Economic Analysis . 2949605 . Chairat Aemkulwat II. Testing for Serial Correlation

• Step 3: Regress t on unem and t‐101 = 2.157 ‐.3929unem + .6449resid01(‐1){t} {5.247}

n=48, R2=.380747

• 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.

29

Example: Static Phillips Revisited

I. Properties II. Testing III. Remedial IV. ARCH V. Hetero & S.C.14. Serial Correlation. Quantitative Methods of Economic Analysis . 2949605 . Chairat Aemkulwat II. Testing for Serial Correlation



Sample: 1 49

Included observations: 49


C 1.42361 1.719015 0.828154 0.4118

UNEM 0.467626 0.289126 1.617376 0.1125







Step 1: Regress inf on unemStep 2: Eviews: Proc/Make Residual Series (named resid01)


Dependent Variable: RESID01





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

R-squared 0.380747 Mean dependent var -0.10207






Step 3: Regress t on unem and t‐1 to find in the case that unem is not strictly exogenous.


Breusch-Godfrey Serial Correlation LM Test:

F-statistic 27.83291 Probability 0.000003

Obs*R-squared 18.47161 Probability 0.000017

Test Equation:

Dependent Variable: RESID

Presample missing value lagged residuals set to zero.


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


Eviews : the case of nonstrictly exogeneous regressorsIn 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.


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 hypothesisH0: 1= 2= ... = q= 0

• We can run the regression of t on xt1, .... , xtk , ut‐1 , .... , ut‐q

• Compute the F test for joint significance of ut‐1 , .... , ut‐q or the LM statistic is

LM = (n‐q)R2u^ ~ 2qThis is called the Breusch‐Godfrey test for AR(q) serial correlation


Example: Expectations Augmented Phillips Curve Revisited

• Regress D(inf) on unem

t = 3.03 ‐ 0.543umemtn = 48 R2 = 0.108 R2 bar= 0.088

• Suppose we wish to test H0: 1= 2= 0 in the AR(2) modelut = 1ut‐1 + 2ut‐2+ unem + et

• Run the regression ofresid c unem resid(‐1) resid(‐2)


Eviews: Expectations Augmented Phillips Curve Revisited

• 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.

• H0: 1= 2= 0F‐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)!


Breusch-Godfrey Serial Correlation LM Test:F-statistic 4.408984 Probability 0.017979

Obs*R-squared 8.013607 Probability 0.018191Test Equation:Dependent Variable: RESIDMethod: Least SquaresPresample missing value lagged residuals set to zero.

Variable Coefficient Std. Error t-Statistic Prob. C -0.83592 1.31912 -0.633697 0.5296

UNEM 0.144109 0.220873 0.652453 0.5175RESID(-1) -0.05971 0.138511 -0.431046 0.6685RESID(-2) -0.4168 0.140903 -2.958062 0.005

R-squared 0.16695 Mean dependent var 4.39E-16

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.


III. Correcting for serial correlation

Remedial measures: After detecting serial correlation, we now learn how to fix the problem.

1) Assume strictly exogenous regressors1.1 Known : find generalized least squares (GLS) estimators1.2 Use estimated ‐ Feasible GLS (FGLS)

• Cochrance‐Orcutt Method• Iterated Cochrance‐Orcutt Method• Differencing

2) Assume without strictly exogeneity– find SC‐robust standard error (heteroskedasticity and autocorrelation

consistent standard error)

37I. Properties II. Testing III. Remedial IV. ARCH V. Hetero & S.C.14. Serial Correlation. Quantitative Methods of Economic Analysis . 2949605 . Chairat Aemkulwat III. Correcting for serial correlation

1.1) Find best linear unbiased estimators (BLUE) in the AR(1) model

• Assume the errors {ut} follow the AR(1) model:

ut = ut‐1 + et

where et ~ i.i.d(0, e2) and || < 1

• Consider the simple regression model

yt= 0 + 1xt + ut

Note that Var(ut)= e2 /(1‐ 2)


Find First Observation in GLS Estimation

• We can show that for t2,= 0 + 1 + et

where quasi‐differenced data (|| < 1) are = yt ‐ yt‐1; = 1 ‐ ; = xt1 ‐ xt‐1,1

• Equation for t=1 isy1= 0 + 1x1 + u1 (*)

Using the fact that Var(ut)= e2 /(1‐ 2), we multiply (*) by (1‐ 2)1/2 to

get errors with the same variance (e2), i.e.,

(1‐ 2)1/2 y1= (1‐ 2)1/2 0 + 1 (1‐ 2)1/2 x1 + (1‐ 2)1/2u1


• For the multiple regression model, we can easily find the generalized difference equation for j=2,....,k:

= 0 + 1 + ...+ k + error

where = yt ‐ yt‐1; = 1 ‐ ; = xtj ‐ xt‐1,j

Generalized difference equation


GLS estimators are BLUE


Since the errors {et} are assumed to be an i.i.d. sequence with mean zero and constant variance.

E(et) = 0var(et) = e

2

E(etet‐1) = 0

GLS estimators are BLUE since the errors in the transformed equation are serially uncorrelated and homoskedastic.Thus, t and F statistics are valid

14. Serial Correlation. Quantitative Methods of Economic Analysis . 2949605 . Chairat Aemkulwat III. Correcting for serial correlation

1.2. FGLS Estimation with AR(1) errors

• The “” is a population parameter. We rarely know the true value of in practice

• We know how to get the a consistent estimator of . That is , run the regression of

t on t‐1

to obtain the sample counterpart


FGLS Estimation with AR(1) errors

We now can run the model with :

= 0 + 1 + ...+ k + error= yt - yt-1 = 1 - = xtj - xt-1,j j=2,....,k

This results in the feasible GLS (FGLS) estimator of the j.

FGLS estimators are not unbiased, but consistent.

FGLS estimators are asymptotically more efficient than the OLS estimator when {ut} is the AR(1) process.


OLS-Assumptions FGLS-Assumptions

1 Cov(xt,ut)=0 Cov(xt,ut)=02 - Cov(xt-1+xt+1,ut)=0

Cochrance-Orcutt Estimation

Comparing OLS and FGLSyt = 0 + 1xt + ut


CO and PW Estimation

• If we omit the first observation, FGLS estimation is called the Cochrance‐Orcutt (CO) estimation.

• If we include the first observation, FGLS estimation is called the Prais‐Winsten (PW) estimation.


• Step 1: OLS output

= 1.423+ .468unem

Since = .573 and t = 4.98 (t>2), there is very strong evidence of serial correlation.

• Step 2: Cochrance‐Orcutt Estimation: = .573

inft‐ inft‐1= (1‐ ) 0 + 1(unem‐ unemt‐1) + et

inft ‐ .573inft‐1= 5.51 – 0.28(unem‐.573unemt‐1)(s.e.) (2.04) (.32)

n=48 R2 = .016


Example: Static Phillips Model


Dependent Variable: INF-.572735*INF(-1)

Method: Least SquaresSample(adjusted): 2 49Included observations: 48 after adjusting endpoints

Variable Coefficient Std. Error t-Statistic Prob. 1-.572735 5.509824 2.037318 2.70445 0.0096

UNEM-.572735*UNEM(-1) -0.27931 0.321911 -0.867665 0.3901R-squared 0.016103 Mean dependent var 1.658889Adjusted R-squared -0.00529 S.D. dependent var 2.349715S.E. of regression 2.355918 Akaike info criterion 4.592512Sum squared resid 255.316 Schwarz criterion 4.670478Log likelihood -108.22 Durbin-Watson stat 1.217816

Cochrane‐Orcutt: =.573


1. There is inflation‐unemployment tradeoff.

2. s.e.’s under the CO method are uniformly higher.

3. The CO method reports one fewer observation than OLS.

4. R‐squared in the OLS is valid whereas R‐squared in CO estimation is not meaningful.

Note that 5.51 =(1‐ ) = (1‐.573) . Thus, = 12.9

Coefficient OLS Cochrane-Orcutt=.573

Intercept 1.42 5.51(s.e.) (1.72) (2.04){t} {.83} {2.71}

Unem 0.47 -0.28(s.e.) (0.289) (0.32){t} {1.62} {-0.869}

(Observations) 49 48R-Squared 0.0527 0.016

OLS and Cochrane‐Orcutt Compared


Iterative Cochrance‐Orcutt Method

• In practice, the Cochrance‐Orcutt method is used in an iterative Scheme

• Steps for iterative Cochrance‐Orcutt MethodStep 1: use to find run the following regression.

yt ‐ yt‐1 = 0(1 ‐ ) + 1(xt1 ‐ xt‐1,1) + etStep 2: We can obtain a new set of residuals and a new estimate

of .Step 3: Repeat step (1) again until the estimate of changes by

very little.• By iterative method above, we can find that =.774 for the static Phillips model.

• Eviews: Regress the following: y c x1 and AR(1). The coefficient of AR(1) is the iterative Cochrance‐Orcutt estimate of





Estimation settings: tol= 0.00010, derivs=accurate mixed (linear)

Convergence achieved after 6 iterations


C 7.583459 2.531239 2.995947 0.0044

UNEM -0.66534 0.361515 -1.840412 0.0723

AR(1) 0.774051 0.104367 7.416633 0






Durbin-Watson stat 1.593634 Prob(F-statistic) 0

Inverted AR Roots 0.77

Iterated Cochrane‐Orcutt: =.774Obtained by regressing in Eviews: inf c unem ar(1)


Dependent Variable: INF-.774*INF(-1)Method: Least SquaresSample(adjusted): 2 49Included observations: 48 after adjusting endpointsVariable Coefficient Std. Error t-Statistic Prob. 1-.774 7.58309 2.380327 3.185734 0.0026UNEM-.774*UNEM(-1) -0.66526 0.319605 -2.081503 0.043R-squared 0.08608 Mean dependent var 0.827412Adjusted R-squared 0.066213 S.D. dependent var 2.356872S.E. of regression 2.277508 Akaike info criterion 4.524815Sum squared resid 238.604 Schwarz criterion 4.602781Log likelihood -106.596 Durbin-Watson stat 1.593539

Iterated Cochrane‐Orcutt: =.774inf‐.774*inf(‐1) (1‐.774) unem‐.774*unem(‐1)


Coefficent OLS Cochrane-Orcutt

Iterated Cochrane-

Orcutt

=.573 =.774

Intercept 1.42 5.51 -7.58(s.e.) (1.72) (2.04) (2.38){t} {.83} {2.71} {3.19}

Unem 0.47 -0.28 -0.665

(s.e.) (0.289) (0.32) (0.319){t} {1.62} {-0.869} {-2.08}

(Observations) 49 48 48

R-Squared 0.052 0.016 0.086

1. Negative Relationship between inflation and unemployment.

2. R‐squared

3. Observations

4 Note that (1‐ ) = ‐7.58 (1‐.774) = ‐7.58 Thus, = ‐33.5

OLS and Iterated Cochrance‐OrcuttCompared


2. Differencing and Serial Correlation

• One benefit of differencing: It is a transformation for making an integrated process weakly dependent [I(0)].


yt= 0 + 1xt + ut

• We have learned as follows.1) If ytor xt are integrated of order one I(1), OLS test statistics can be misleading.2) Differencing can mitigate or may get rid of serially

correlated errors.


Differencing and Serial Correlation

• Differencing results inyt= 0 + 1xt + utyt= 1xt + etet = ut = ut ‐ ut‐1

Assume {et} follow an i.i.d sequence with mean 0 and variance e2.

• Differencing in Phillips modelConsider the first difference of variables in the Phillips Model

inft – inft‐1 = 1(unemt – unemt‐1) + etinft – inft‐1 : unanticipated inflationunemt – unemt‐1 : unanticipated unemploymentet : error term (supply shock)


Differencing in Phillips model

• Do differencing the static Phillips model and estimate.inft –inft‐1 = ‐.078 ‐0.84(unemt‐unemt‐1)[t‐stat] [‐.224] [‐2.68]n=48, R2=.35046, R2‐bar=.116243

• Results:1. There is tradeoff between inflation and unemployment.2. = ‐.03047 (p‐value= 0.8444); there is no evidence of serial correlation after differencing.

3. Note that inf and unemmay contain a unit root (how do we know?), so differencing might be needed to produce I(0) variables.


Dependent Variable: D(INF)

Method: Least SquaresSample(adjusted): 2 49Included observations: 48 after adjusting endpoints

Variable Coefficient Std. Error t-Statistic Prob. C -0.07818 0.348462 -0.224351 0.8235

D(UNEM) -0.84217 0.314251 -2.679931 0.0102R-squared 0.135046 Mean dependent var -0.10625Adjusted R-squared 0.116243 S.D. dependent var 2.566926S.E. of regression 2.413125 Akaike info criterion 4.640496Sum squared resid 267.8659 Schwarz criterion 4.718463Log likelihood -109.372 F-statistic 7.18203Durbin-Watson stat 1.849401 Prob(F-statistic) 0.010184

Results from differencing:1. There is a tradeoff between inflation and unemployment.


Breusch-Godfrey Serial Correlation LM Test:

F-statistic 0.038992 Probability 0.844353

Obs*R-squared 0.041556 Probability 0.838469

Test Equation:

Dependent Variable: RESID


Presample missing value lagged residuals set to zero.


C -0.00063 0.352175 -0.001793 0.9986

D(UNEM) 0.016852 0.328852 0.051245 0.9594

RESID(-1) -0.03047 0.154296 -0.197465 0.8444

R-squared 0.000866 Mean dependent var 1.13E-17


2. = ‐.03047 (p‐value= 0.8444); there is no evidence of serial correlation after differencing.Eviews: View/Residual Tests/Serial Correlation LM Test of AR(1)


3. Serial Correlation‐Robust Inference After OLS

• Reasons for taking this approach:#1. Explanatory variables may not be strictly exogeneous;

thus, FGLS is not even consistent.#2. In most application of FGLS, the errors are assumed to

follow AR(1). It may be better to compute standard errors for the OLS estimates that are robust to more general forms of serial correlation.

• Given a modelyt = 0 + 1xt1+ 2xt2+ .... + kxtk + ut


Model • Given a model

yt = 0 + 1xt1+ 2xt2+ .... + kxtk + utut = yt – 0 – 1xt1 – .... – kxtk

• se( ) = Serial correlation‐robust standard error for se( ) = [“se( )”/ 2

wherethe standard error of the regression

“se( )” se( ) in the original equationWe don’t know


Find se( ) = [“se( )”/


• Write auxiliary regression for xt1 as xt1 = 0 + 2xt2+ .... + kxtk + r1tr1t = xt1 – 0 – 2xt2 – .... – kxtk

• = 1t• g = 4(n/100)2/9 (Newley West and Eviews)

• Annual: g=1,2• Quarter: g=4,8• Monthly: g=12,24


Coefficient OLS s.e. SC s.e prob SC probintercept 1.42361 1.719015 1.515019 0.4118 0.3522UNEM 0.467626 0.289126 0.291606 0.1125 0.1155


Steps in Eviews

• Step 1: Obtain the original output.

= 1.423+ .468unem

• Step 2: In the outputChoose "Estimate"

– Click Options√ Heteroskedasticity Consistent Coefficient Covariance

– choose Newley‐West


Dependent Variable: INFMethod: Least SquaresSample: 1 49Included observations: 49

Variable Coefficient Std. Error t-Statistic Prob. C 1.42361 1.719015 0.828154 0.4118

UNEM 0.467626 0.289126 1.617376 0.1125R-squared 0.052723 Mean dependent var 4.108163Adjusted R-squared 0.032568 S.D. dependent var 3.182821S.E. of regression 3.130562 Akaike info criterion 5.160262Sum squared resid 460.6198 Schwarz criterion 5.237479Log likelihood -124.426 F-statistic 2.615904Durbin-Watson stat 0.8027 Prob(F-statistic) 0.11249

Step 1: Regress inf on unem


Dependent Variable: INFMethod: Least SquaresSample: 1 49Included observations: 49

Newey-West HAC Standard Errors & Covariance (lag truncation=3)Variable Coefficient Std. Error t-Statistic Prob. C 1.42361 1.515019 0.939665 0.3522UNEM 0.467626 0.291606 1.603624 0.1155R-squared 0.052723 Mean dependent var 4.108163Adjusted R-squared 0.032568 S.D. dependent var 3.182821S.E. of regression 3.130562 Akaike info criterion 5.160262Sum squared resid 460.6198 Schwarz criterion 5.237479Log likelihood -124.4264 F-statistic 2.615904Durbin-Watson stat 0.8027 Prob(F-statistic) 0.11249

Step 2: In the outputChoose "Estimate"

Click Options√ Heteroskedasticity Consistent Coefficient Covariance

choose Newley-West


IV. ARCH model

• We are interested in the dynamic forms of heteroskedasticity.

• Given a model,

yt = 0 + 1zt + ut

• Suppose Gauss Markov assumptions hold. 1. OLS have desirable properties.2. TS.4 implies that E(ut2|Z) = constant. (Z is all n outcomes of zt)3. Even when variance of ut given Z is constant, there are other

ways that heteroskedasticiy can arise.

65I. Properties II. Testing III. Remedial IV. ARCH V. Hetero & S.C.14. Serial Correlation. Quantitative Methods of Economic Analysis . 2949605 . Chairat Aemkulwat IV. ARCH model

OLS estimators and ARCH

Engle (1982) suggested the conditional variance of ut depends on past errors in the form,

E(ut2|ut‐1, ut‐2,…) = E(ut2|ut‐1) = 0 + 1ut‐12

This is called the first‐order ARCH model. Note that 1≠0 implies that there is dynamics in the variance equation.

Example – UK inflationEngle found that a larger magnitude of the error variance in the previous time period was associated with a larger error variance in the current period.


ARCH Model

We can write the equation in the form

ut2 = 0 + 1ut‐12 + vt

Let E(vt|ut‐1, ut‐2 ,…)=0 and 1<1 for stability condition.

Why ARCH model?1) It is possible to obtain consistent estimators that are asymptotically more efficient than OLS. A WLS procedure will do the trick.2) It gives an insight into the dynamics in the conditional variance.


Example: ARCH in Stock Return

• = 0.180 + 0.059returnt‐1[t] [2.22] [1.55]

n = 689, R2 = .0035, R2bar = .0020

• The heteroskedasticity can be characterized by ARCH model.t2 = 2.95 + .337 t‐1

2

[t] [6.766] [9.37]

• Notes:1) Note that there is no serial correlation: = .0014 (t=0.038).2) The t‐statistic (t=‐9.4) is very high, indicating strong ARCH.3) Implications: a larger variance in stock last period implies a

larger variance in stock this period.


Dependent Variable: RESID01^2





C 2.947433 0.440234 6.695148 0

RESID01(-1)^2 0.337062 0.035947 9.376707 0







Heteroskedasticity (ARCH model):t2 = 2.95 + .337 t‐1

2


V. Heteroskedasticity and Serial Correlation

• Suppose TS.1‐TS.3 hold. Consider a modelyt = 0 + 1xt1 + 2xt2 + ... + kxtk + utut = vtvt = vt‐1 + et , <1

where 1) {et} i.i.d. (0, 2

e)2) E(etX) = 03) ht = f(xij)

• Thus,Var(ut) = 2

v ht2

v =2e/(1‐ 2)

70I. Properties II. Testing III. Remedial IV. ARCH V. Hetero & S.C.14. Serial Correlation. Quantitative Methods of Economic Analysis . 2949605 . Chairat Aemkulwat V. Heteroskedasticity and Serial Correlation

Steps: FGLS with hetero and AR(1) SC

Step 1: Run the regression of yt on xt1, xt2 , ..., xtk and obtain residuals t

Step 2: Regress log( t2) on xt1, xt2 , ..., xtk (or t and t

2) and obtain fitted values, call t.

Step 3: obtain t by

t = exp( t)

Step 4: Estimate the transformed equation

yt/ t = 0/ t + 1xt1/ t + ... + kxtk/ t + vtStep 5: Correct serial correlation by standard Cochrance‐Orcutt or

Prais‐Winsten methods.


Heteroskedasticity in Time Series Regressions

• Suppose TS.1’‐TS.5’ hold, the usual OLS statistics are asymptotically valid. – TS.3’ implies that model misspecifications are ruled out.– TS.4’ and TS.5’ rule out heteroskedasticity and serial correlation.

• Serial Correlated Errors – more pressing in Time SeriesHeteroskedastic errors – more pressing in Cross Section.


Testing for Heteroskedasticity

• However, heteroskedasticity can also occur in time series regression models. With heteroskedasticity (TS.1‐TS.3 and TS.5 are true),– are unbiased and consistent, but– se( ) , t‐stat, and F‐stat are no longer valid, even n is large.

• If we assume that ut are serially uncorrelated, we could test for heteroskedasticity by– Breusch‐Pagan Test– White Test

Suppose TS.1’‐TS.3’ hold. Consider the k‐regressor model,yt = 0 + 1xt1 + 2xt2 + ... + kxtk + ut


Steps in correcting for heteroskedasticity in time series

• Step 1: Test for serial correlation and correct by, say, the iterative Cochrance‐Orcutt method.

• Step 2: Test for heteroskedasticity, for example, Breusch‐Pagan Test.

ut2= 0 + 1x1t + 2x2t +...+ kxkt + vtwhere the null hypothesis is H0 : 1 = 2 =… = k = 0

• Step 3: if heteroskedasticity is found, to correct we may• use the heteroskedasticity‐robust procedures.• use weighted least squares method.


Example: Heteroskedasticity in the EM Hypothesis

• t = 0.180 + 0.059returnt‐1 {t {2.224} {1.549}

n = 689, R2 = .0035,R2bar = .0020With such a large sample, there is no evidence against EMH. Note that there is no serial correlation: =.0014 (t=0.038).

• EMH states that the expected return given past information should be constant. It says nothing about the conditional variance. Breusch‐Pagan Test suggests

t2 = 4.66 – 1.104returnt‐1

[t] [10.89] [‐5.48]• There is a strong evidence of heteroskedasticity.


Example: Heteroskedasticityin the EM Hypothesis

• t = 0.180 + 0.059returnt‐1 (*) t2 = 4.66 – 1.104returnt‐1 (**)

• Notes1. In (**), the coefficient of return(‐1) is negative. This

implies that the volatility in stock returns is lower when the previous return is high, and vice versa.

2. In short, statistically, the expected value of stock returns does not depend on past returns shown in (*), but the variance of returns does depend on past returns shown in (**).



Variable Coefficient Std. Error t-Statistic Prob. RESID01(-1) 0.001405 0.03815 0.036828 0.9706

R-squared 0.000002 Mean dependent var -0.00117Adjusted R-squared 0.000002 S.D. dependent var 2.110171S.E. of regression 2.110169 Akaike info criterion 4.332865Sum squared resid 3059.082 Schwarz criterion 4.339455Log likelihood -1489.51 Durbin-Watson stat 1.999909

Stock Return: No Serial Correlation:


Dependent Variable: RESID01^2





C 4.656501 0.427679 10.88784 0

RETURN(-1) -1.10413 0.201403 -5.482207 0







Breusch‐Pagan Test:t2 = 4.66 – 1.104returnt‐1


Recap on Serial Correlation

• Properties of OLS with Serially Correlated Errors• Testing for Serial Correlation• Correcting for Serial Correlation• Autoregressive conditional heteroskedasticity

(ARCH) model• Heteroskedasticity in Time Series


80

Good Luck!on Your Final Examination


14. time series analysis: i. ols serial correlation ii. iii. iv....

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