becin 002 chapter 6 · used test for serial correlation, there are limitations: 1. the test is for...

Chapter 6Chapter 6

Autocorrelation or Serial Correlation

Section 6.1Section 6.1

Introduction

2

Evaluating Econometric WorkHow does an analyst know when the econometric work is completed?

3

Evaluating Econometric WorkEconometric Results – indeed a term that for many economists conjures up horrifying visions of well-meaning but perhaps marginally skilled, likely nocturnal, individuals sorting through endless piles of computer print-outs. One “final” print-out is then chosen for seemingly mysterious reasons and the rest discarded to be recycled through a local paper processor and another computer printer for other “econometricians” to repeat the process ad infinitum. Besides supplying a lucrative business for the paper recyclers, what useful output, if any, results from such a process? This question lies at the heart of the so called “science of econometrics as currently applied,” a practice which has been called “data-mining”, “number crunching”, “model sifting”, “data grubbing”, “fishing”, “data messaging”, and even “alchemy” among other less palatable terms. All of these euphemisms describe basically, the same process: choosing an econometric model based on repeated experimentation with available sample data.

- Ziemer (1984)4

Evaluating Econometric Work“Econometrics may not have the everlasting charm of Holmesian characters and adventures, or even a famous resident of Baker Street, but there is much in his methodological approach to the solving of criminal cases that is of relevance to applied econometric modeling. Holmesian detection may be interpreted as Holmesian detection may be interpreted as accommodating the relationship between data theory, modeling procedures, deductions and inferences, analysis of biases, testing of theories, re-evaluation and reformulation of theories, and finally reaching a solution to the problem at hand. With this in mind, can applied econometricians learn anything from the master of detection?

- McAleer (1994)5

Key Diagnostics

Diagnostic Component of Econometric Model Under Consideration

Serial Correlation (Autocorrelation)

Error (or Disturbance) Term

Heteroscedasticity Error (or Disturbance) Term

Collinearity Diagnostics Explanatory Variables

6

Collinearity Diagnostics Explanatory Variables

Influence diagnostics Observations

Structural Change Structural Parameters

�� See Beggs (1988).


Autocorrelation or

Serial Correlation

Autocorrelation or Serial Correlation� Definition� Consequences� Formal Tests

– Durbin-Watson Test– Nonparametric Runs Test– Durbin's h-Test– Durbin's m-Test– Lagrange Multiplier (LM) Test– Box-Pierce Test (Q Statistic)– Ljung-Box Test (Q* Statistic)– (Small-sample modification of Box-Pierce Q

Statistic)� Solution

– Generalized Least Squares88

Formal Definition of Autocorrelation or Serial CorrelationAutocorrelation or serial correlation refers to the lack of independence of error (or disturbance) terms. Autocorrelation and serial correlation refer to the same phenomenon.Simply put, a systematic pattern exists in the residuals of the econometric model. Ideally, the residuals, which represent a composite of all factors not embedded in the model, should exhibit no pattern.That is to say, the residuals should follow a white-noise (or random) pattern.

99

Prevalence of Serial CorrelationWith the use of time-series data in econometric applications, serial correlation is “public enemy number one.” Systematic patterns in the error terms commonly arise due to the (inadvertent) omission of explanatory variables in econometric models. These variables might come from disciplines other than economics, finance, or business; for example, psychology and sociology.business; for example, psychology and sociology.Alternatively, these variables might represent factors that simply are difficult to quantify, such as tastes and preferences of consumers or technological innovation on the part of producers.

1010

Consequences of Serial Correlation� Bishop (1981)� Errors “contaminated” with autocorrelation or serial

correlationPotential of discovering “spurious” relationships due to problems with autocorrelated errors (Granger and Newbold, 1974)

� Difficulties with structural analysis and forecastingIf the error structure is autoregressive, then OLS estimates of the regression parameters are (1) unbiased, (2) consistent, but (3) inefficient in small and in large samples.

1111 continued...

� The estimates of the standard errors of the coefficients in any econometric model are biased downward if the residuals are positively autocorrelated. They are biased upward if the residuals are negatively autocorrelated.

� Therefore, the calculated t statistic is biased upward or downward in the opposite direction of the bias in the

Consequences of Serial Correlation

downward in the opposite direction of the bias in the estimated standard error of that coefficient.

� Granger and Newbold (1974) further suggest that the econometric results can be defined as “nonsense” if R2 >DW(d).

1212 continued...

Consequences of Serial CorrelationPositive autocorrelation of the errors generally tends to make the estimate of the error variance too small, so confidence intervals are too narrow and null hypotheses are rejected with a higher probability than the stated significance level.Negative autocorrelation of the errors generally tends to make the estimate of the error variance too large, so make the estimate of the error variance too large, so confidence intervals are too wide; also the power of significance tests is reduced.With either positive or negative autocorrelation, least-squares parameter estimates usually are not as efficient as generalized least-squares parameter estimates.

1313

Regression with Autocorrelated Errors� Ordinary regression analysis is based on several statistical

assumptions. One key assumption is that the errors are independent of each other. However, with time series data, the ordinary regression residuals usually are correlated over time.

� Violation of the independent errors assumption has three important consequences for ordinary regression.important consequences for ordinary regression.– First, statistical tests of the significance of the

parameters and the confidence limits for the predicted values are not correct.

– Second, the estimates of the regression coefficients are not as efficient as they would be if the autocorrelation were taken into account.

– Third, because the ordinary regression residuals are not independent, they contain information that can be used to improve the prediction of future values.1414

Solution to the Serial Correlation ProblemGeneralized Least Squares (GLS)The AUTOREG procedure solves this problem by augmenting the regression model with an autoregressive model for the random error, thereby accounting for the systematic pattern of the errors. Instead of the usual regression model, the following autoregressive error model is used:

εβxy ttt +′=

The notation indicates that each vt is normally and independently distributed with mean 0 and variance σ2.

1515

),0( ~

...

2

2211

σ

εφεφεφε

εβ

INv

v

xy

t

tmtmttt

ttt

+−−−−=

+′=

−−−

),0( ~ 2σINvt

continued...

Solution to the Serial Correlation ProblemGeneralized Least Squares (GLS)By simultaneously estimating the regression coefficients β and the autoregressive error model parameters φi , the AUTOREG procedure corrects the regression estimates for autocorrelation. Thus, this kind of regression analysis is often called autoregressive error correction or serial correlation correction.correlation correction.

This technique is also called the use of generalized least squares (GLS).

1616

Predicted Values and ResidualsThe AUTOREG procedure can produce two kinds of predicted values and corresponding residuals and confidence limits.

� The first kind of predicted value is obtained from only the structural part of the model. This predicted value is an estimate of the unconditional mean of the dependent variable at time t.dependent variable at time t.

� The second kind of predicted value includes both the structural part of the model and the predicted value of the autoregressive error process.

Both the structural part and autoregressive error process of the model (termed the full model) are used to forecast future values.

1717 continued...

Predicted Values and ResidualsUse the OUTPUT statement to store predicted values and residuals in a SAS data set and to output other values such as confidence limits and variance estimates.

� The P= option specifies an output variable to contain the full model predicted values.

� The PM= option names an output variable for the predicted (unconditional) mean.

� The R= and RM= options specify output variables for the corresponding residuals, computed as the actual value minus the predicted value.

1818

Serial Correlation� Disturbance terms are not independent.

� The correlation between εt and εt-k is called an autocorrelation of order k.

jiE

TiXXXY

ji

tktkttt

≠≠=+++++=

0)(

,...,2,1...22110εε

εββββ

t t-kautocorrelation of order k.

1919 continued...

Serial CorrelationRecommend a graphical analysis of plotting the residuals over time to determine the existence of a non-random or systematic pattern.

Time Positive Correlation

Residuals

2020 ...

Time Negative Correlation

Residuals

Section 6.3 Section 6.3

Tests for Serial Correlation

The Durbin-Watson TestAR(1) process

or d statistic

ttt v+= −1ρεε

0:

0:

1

0

≠=

ρρ

H

H

4ˆ

)ˆˆ(0

2

2

21

≤−

=≤∑

∑=

−

n

n

ttt

e

eeDW

22

4d then -1, if

0;d then 1, if 2;d then 0, If

======

ρρρ

continued...

∑

∑

=

=−

≈−=−≈n

tt

n

ttt

e

eedDW

dDW

1

2

21ˆˆ

2

)(1ˆ)1(2)( ρρ that so

ˆ1

2∑

=tte

� dL,dU depend on α, k, n.� DW is invalid with models that contain no intercept and

models that contain lagged dependent variables.� The distribution of DW(d) is reported by Durbin and

Watson (1950, 1951).

The Durbin-Watson Test

2323 continued...

f(d)Reject H0

Evidence of positiveautocorrelation

Reject H0

Evidence of negativeautocorrelation

Zone of indecision

Zone of indecision

continued...

dL 4-dL2 4-dUdU 40 d

Accept H0

24

The Durbin-Watson Test� The sampling distribution of d depends on the values

of the exogenous variables and hence Durbin and Watson derived upper (dU) limits and lower (dL) limits for the significance levels for d.

� Tables of the distribution are found in most econometric textbooks.

� The Durbin-Watson test perhaps is the most used procedure in econometric applications.

2525

The Durbin-Watson Statistic

2626 continued...

Appendix G: Statistical Table

27

Limitations of the Durbin-Watson TestAlthough the Durbin-Watson test is the most commonly used test for serial correlation, there are limitations:1. The test is for first-order serial correlation only.2. The test might be inconclusive.3. The test cannot be applied in models with lagged

dependent variables.dependent variables.4. The test cannot be applied in models without

intercepts.

2828

Additional TablesThere are other tables for the DW test that have been prepared to take care of special situations. Some of these are:

1. R.W. Farebrother (1980) provides tables for regression models with no intercept term.

2. Savin and White (1977) present tables for the DW test 2. Savin and White (1977) present tables for the DW test for samples with 6 to 200 observations and for as many as 20 regressors.

2929 continued...

Additional Tables3. Wallis (1972) gives tables for regression models with quarterly

data. Here you want to test for fourth-order autocorrelation rather than the first-order autocorrelation. In this case, the DW statistic is

∑

∑=

−−=

n2t

n

5t

24tt

4

û

)ûû(d

Wallis provides 5% critical values dL and dU for two situations: where the k regressors include an intercept (but not a full set of seasonal dummy variables) and another where the regressors include four quarterly seasonal dummy variables. In each case the critical values are for testing H0: ρ =0 against H1: ρ > 0. For the hypothesis H1: ρ < 0, Wallis suggests that the appropriate critical values are (4-dU) and (4-dL). King and Giles (1978) give further significance points for Wallis tests.

3030 continued...

∑=1t

tu

Additional Tables4. King (1981) gives the 5% points for dL and dU quarterly

time-series data with trend and/or seasonal dummy variables. These tables are for testing first-order autocorrelation.

5. King (1983) gives tables for the DW test for monthly data. In case of monthly data, you want to test for twelfth-order autocorrelation.twelfth-order autocorrelation.

3131

Nonparametric Runs Test (Gujarti, 1978)� More general than the DW test. Interest in H0: ρ = 0.

– Test of AR(1) process in the error termsN+ = number of positive residualsN- = number of negative residualsN = number of observationsNr = number of runs

� Example:

� Test Statistic:

� Reject H0 (non-autocorrelation) if the test statistic is too large in absolute value.

3232

[ ][ ] ))1(/())2(2(

/)2(2 −−=

=−+−+

−+

NNNNNNNNVAR

NNNNE

r

r

)1,0(N)N(VAR/))N(EN(Z rrr ≈−=

The REG Procedure

Model: MODEL1

Dependent Variable: lnpcg

Number of Observations Read 36

Number of Observations Used 36

Analysis of Variance

Sum of Mean

Source DF Squares Square F Value Pr > F

3333


Model 6 0.78035 0.13006 150.85

Parameter Estimates

Parameter Standard

Variable DF Estimate Error t Value Pr > |t|

Intercept 1 -17.27084 1.71977 -10.04

The REG Procedure

Model: MODEL1


Durbin-Watson D 0.786

Pr < DW DW 1.0000

Number of Observations 36

1st Order Autocorrelation 0.601

NOTE: PrDW is the p-value for testing negative autocorrelation.

3535

In this example, sample evidence exists to suggest the presence of positive serial correlation, which is t he more common form of pattern in the residuals in regard t o the use of economic or financial data.

Obs year lnpcgres

19 1978 0.024483

20 1979 -0.007387

21 1980 -0.039505

22 1981 -0.014226

23 1982 0.022968

24 1983 0.042105

25 1984 -0.011793

26 1985 -0.028349

Obs year lnpcgres

1 1960 0.017118

2 1961 0.010675

3 1962 0.019701

4 1963 0.025370

5 1964 -0.019348

6 1965 -0.043199

7 1966 -0.045077

8 1967 -0.056039 26 1985 -0.028349

27 1986 -0.007728

28 1987 0.016606

29 1988 -0.019070

30 1989 -0.005426

31 1990 -0.008096

32 1991 -0.025680

33 1992 -0.015855

34 1993 0.013422

35 1994 0.007171

36 1995 0.001383 3636

8 1967 -0.056039

9 1968 -0.032779

10 1969 0.003223

11 1970 0.008482

12 1971 0.021400

13 1972 0.016612

14 1973 -0.015336

15 1974 0.006500

16 1975 0.039297

17 1976 0.048282

18 1977 0.050095

The Greene ProblemIn the Greene problem for gasoline, DW = 0.786 and = 0.601.

Use of Nonparametric Runs testN = 36 N+ = 19 N- = 17 Nr = 11

ρ̂

[ ][ ] 69.8))1(/())2(2(

94.17/)2(2 =−−=

==−+−+

−+

NNNNNNNNVAR

NNNNE

r

r

3737

[ ] 69.8))1(/())2(2( =−−= NNNNNNNNVAR r

.05 at

reject at

====

−=−=−=

−=

α96.1005

35.295.2

94.6

69.8

94.1711

)())((

0

crit

rrr

Z

.:ρ H, .α

Z

NVARNENZ

Analysis Limitations� Analysts must recognize that a “good” Durbin-Watson

statistic is insufficient evidence upon which to conclude that the error structure is “contamination free” in terms of autocorrelation. The Durbin-Watson test is only applicable for the presence of first-order autocorrelation.

� There is little reason to suppose that the correct model for residuals is AR(1). A mixed, autoregressive, moving-residuals is AR(1). A mixed, autoregressive, moving-average (ARMA) structure is much more likely to be correct, especially with quarterly, monthly, and weekly frequencies of time-series data. Modeling of the residuals can be employed following the methodology of Box and Jenkins (1976).

� Owing to higher frequencies of time-series data used in applied econometrics in recent years, the pattern of the error structure generally is more complex than the common AR(1) pattern.

3838

A General Test for Higher-Order Serial CorrelationThe LM Test (Breusch and Pagan, 1980)

� LM - Lagrange multiplier

0...:H

),0(IN~eeu...uuu

.n,...,2,1tuX...Xy

p210

2ttptp2t21t1t

tktkt110t

=ρ==ρ=ρ

σ+ρ++ρ+ρ=

=+β++β+β=

−−−

The Xs might or might not include lagged dependent variables.

1. Estimate by OLS and obtain the least squares residuals.

2. Estimate .

3. Test whether the coefficients of are all zero. Use the conventional F statistic.

3939

0...:H p210 =ρ==ρ=ρ

.ˆ...ˆ1

110 t

p

tiitktktt vuXXu +++++= ∑

=− ργγγ

tû

itû −

Box-Pierce or Ljung-Box Tests� Check the serial correlation pattern of the residuals. You must

be sure that there is no serial correlation (desire white noise).

� H0: no pattern in the residuals (The residuals are white noise.)

� Box and Pierce (1970) suggest looking at not only the first-order autocorrelation but autocorrelation of all orders of residuals.

Calculate where∑=m

krNQ2,Calculate where

is the autocorrelation of lag k, and N is the number of observations in the series.

� If the model fitted is appropriate, .

� Ljung and Box (1978) suggest a modification of the Q statistic for moderate sample sizes.

40

∑=

−−+=m

1k

2k

1r)kN()2N(N*Q

∑=

=k

krNQ1

,

2kr

2~ mQ χ&

Box-Pierce or Ljung-Box TestsWith the Box-Pierce or Ljung-Box tests, you examine the interface of structural models with time-series models.� Use the correlations and partial correlations of the

residuals over time. The idea is to determine the appropriate pattern in the error structure from the autocorrelation and partial autocorrelation functions associated with the residuals.associated with the residuals.

� Autocorrelation functions tell you about moving average (MA) patterns.

� Partial autocorrelation functions tell you about autoregressive (AR) patterns.

� Anticipate ARMA error structures, particularly higher-order AR patterns in residuals of econometric models.

41


Sample Problem:

The Demand for Shrimp

The REG Procedure

Model: MODEL1

Dependent Variable: QSHRIMP




Sum of Mean

43

Sum of Mean


Model 6 2064.71370 344.11895 19.59

Parameter Estimates

Parameter Standard


Intercept 1 29.69939 8.11798 3.66 0.0004

PSHRIMP 1 -0.03223 0.00346 -9.33

The REG Procedure

Model: MODEL1

Dependent Variable: QSHRIMP


Pr < DW 0.6443

Pr > DW 0.3557

The REG Procedure Output

Pr > DW 0.3557


1st Order Autocorrelation -0.049


45

Conclusion: No AR(1) pattern in the residuals

8

12

16

RESID

-8

-4

0

4

10 20 30 40 50 60 70 80 9046

Name of Variable = resQSHRIMP Mean of Working Series -12E-16 Standard Deviation 4.037075 Number of Observations 97

Autocorrelations Lag Cova riance Correlation - 1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 Std Error

The ARIMA Procedure Output

MA(3) Pattern

The autocorrelation function(acf). A plot of the correlation ofthe residuals at various lags.Corr (et, et-k), k = 0, 1, 2, …, 24.

Lag Cova riance Correlation - 1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 Std Error 0 16.297977 1.00000 | |********************| 0 1 -0.803782 -.04932 | . *| . | 0.101535 2 1.365554 0.08379 | . |** . | 0.101781 3 5.065873 0.31083 | . |****** | 0.102 490 4 1.279257 0.07849 | . |** . | 0.111786 5 0.847853 0.05202 | . |* . | 0.112353 6 1.553722 0.09533 | . | ** . | 0.112601 7 2.748583 0.16865 | . |*** . | 0.113 430 8 0.060094 0.00369 | . | . | 0.115 986 9 1.246364 0.07647 | . |** . | 0.115 988 10 2.816294 0.17280 | . |*** . | 0.11 6506

47

Partial Autoco rrelations

Lag Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1

1 -0.04932 | . *| . | 2 0.08155 | . |** . | 3 0.32156 | . |****** | 4 0.12215 | . |** . | 5 0.01467 | . | . | 6 -0.01941 | . | . | 7 0.12325 | . |** . | 8 -0.00334 | . | . | 9 0.02246 | . | . |

AR(3) Pattern

The partial autocorrelation function (PACF).A plot of the 9 0.02246 | . | . |

10 0.09677 | . |** . | 11 0.06154 | . |* . | 12 -0.09759 | . **| . | 13 0.04673 | . |* . | 14 -0.15242 | .***| . | 15 -0.01645 | . | . | 16 -0.19457 | ****| . | 17 0.08137 | . |** . | 18 -0.07893 | . **| . | 19 -0.07409 | . *| . | 20 -0.01932 | . | . |

21 0.06747 | . |* . | 22 -0.19432 | ****| . |

48

A plot of the correlation of the residuals at various lags after netting out intermittent lags.

Partial Autocorrelations Lag Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 23 -0.06713 | . *| . | 24 -0.03305 | . *| . | Autocorrelation Check for White Noise

PROC ARIMA Output

To Chi- Pr > Lag Square DF ChiSq ----------------Autocorrelations------------- 6 12.70 6 0.0480 -0.049 0.084 0.311 0 .078 0.052 0.095 12 20 .18 12 0.0637 0.169 0.004 0.076 0.1 73 0.056 -0.037 18 26.80 18 0.0828 0.159 -0.090 0.001 -0.099 0 .067 -0.093 2 4 36.18 24 0.0527 -0.130 0.065 -0.04 4 -0.218 -0.051 -0.031

49

The Ljung Box Q* statistic reveals that the residual series is not white noise.

Correlogram of

RESID

50

Presence of MA(3), AR(3) pattern

Durbin-Watson Statistics

Order DW Pr < DW Pr > DW

1 2.0921 0.6443 0.3557

2 1.8221 0.1974 0.8026

3 1.3585 0.0007 0.9993

NOTE: PrDW is the p-value for testing negative autocorrelation. and Pr>DW is the p-value for testing negative autocorrelation.

Godfrey's Serial Correlation Test

Alternative LM Pr > LM

AR(1) 0.2799 0.5967

AR(2) 0.9989 0.6069

AR(3) 11.6932 0.0085

51

Presence of AR(3) pattern

Standard Approx


Intercept 1 29.6994 8.1180 3.66 0.0004

PSHRIMP 1 -0.0322 0.003456 -9.33

Breusch-Godfrey Serial Correlation LM Test:

F-statistic 3.975111 Prob. F(3,87) 0.0105 Obs*R-squared 11.69324 Prob. Chi-Square(3) 0.0085

Test Equation: Dependent Variable: RESID Method: Least Squares Sample: 1 97 Included observations: 97 Presample missing value lagged residuals set to zer o.

Coefficient Std. Error t-Statistic Prob.

C -0.064917 7.783149 -0.008341 0.9934 PSHRIMP 0.000324 0.003415 0.094983 0.9245 PSHRIMP 0.000324 0.003415 0.094983 0.9245

PFIN 0.003214 0.013142 0.244517 0.8074 PSHELL1 -0.002645 0.006911 -0.382712 0.7029

ADSHRIMP 0.003714 0.007144 0.519862 0.6045 ADSHELL1 0.002087 0.020725 0.100694 0.9200

ADFIN -0.000244 0.004692 -0.052095 0.9586 RESID(-1) -0.074724 0.109910 -0.679863 0.4984 RESID(-2) 0.091318 0.107147 0.852272 0.3964 RESID(-3) 0.351359 0.106391 3.302520 0.0014

R-squared 0.120549 Mean dependent var -3.13E-15 Adjusted R-squared 0.029571 S.D. dependent var 4.058047 S.E. of regression 3.997597 Akaike info criterion 5.706646 Sum squared resid 1390.328 Schwarz criterion 5.972081 Log likelihood -266.7724 Hannan-Quinn criter. 5.813975 F-statistic 1.325037 Durbin-Watson stat 2.066485 Prob(F-statistic) 0.235832

53

Presence ofAR(3) Pattern

Estimates of Autoregressive Parameters

Standard Lag Coefficient Error t Va lue

1 0.071520 0.101517 0 .70 2 -0.096118 0.101283 -0 .95 3 -0.321561 0.101517 - 3.17 Use of Yule-

Walker

Correcting for serial correlationthrough the use of PROC AUTOREG

Yule -Walker Estimates

SSE 1376.71794 DFE 87 MSE 15.82434 Root MS E 3.97798

SBC 578.680834 AIC 552.933724 Reg ress R-Square 0.5618 Total R-Square 0.6224 Log Likelihood -551.8663 Observations 97

54

Walkerestimates ofφφφφ1, φφφφ2, and φφφφ3

Durbin -Watson Statistics


1 2.0340 0 .5135 0.4865 2 2.0047 0 .5319 0.4681 3 1.8530 0 .2997 0.7003

NOTE: PrDW is the p-value for testing negative autoc orrelation.

Now, no serial correlationexists in the residuals.

55

The AUTOREG Procedure



AR(1) 2.4131 0.1203 AR(2) 2.5993 0.2726 AR(3) 2.6009 0.4573

Standard Approx Variable DF Estimate Error t Value Pr > |t|

GLS Estimates


Intercept 1 29.6938 7.2391 4.10

Partial

Autocorrelations

1 -0.049318

2 0.081553

3 0.321561

Preliminary MSE 14.4802


Starting values of estimates of φφφφ1, φφφφ2, and φφφφ3 in the ML procedure.


Standard

Lag Coefficient Error t Value

1 0.071520 0.101517 0.70

2 -0.096118 0.101283 -0.95

3 -0.321561 0.101517 -3.17

Algorithm converged.

57

Maximum Likelihood Estimates

SSE 1365.10117 DFE 87

MSE 15.69082 Root MSE 3.96116

SBC 578.057305 AIC 552.310196

Regress R-Square 0.5711 Total R-Square 0.6256

Log Likelihood -266.1551 Observations 97

Use of the MaximumLikelihood procedureto produce estimatesof φφφφ1, φφφφ2, and φφφφ3.



1 2.1042 0.6574 0.3426

2 2.0571 0.6368 0.3632

3 1.9762 0.5481 0.4519

NOTE: PrDW is the p-value for use of autoreg procedure testing negative

autocorrelation. 58



AR(1) 2.0021 0.1571

AR(2) 2.2920 0.3179

AR(3) 2.2951 0.5135

Standard Approx


ML Estimates

Intercept 1 30.4400 7.2224 4.21

Autoregressive parameters assumed given.

Standard Approx


Intercept 1 30.4400 7.1700 4.25

Depiction of the Estimated Model for the Qshrimp Problem� The estimated model is based on the Maximum Likelihood

estimates.

� Qshrimpt = 30.4400 - .0341*Pshrimpt - .0138*Pfint + .007794*Pshell1t + .001561*Adshrimpt - .0155Adfint -.003312*Adshell1t + vt

� vt = -.0152*vt-1 + .1256*vt-2 + .3915*vt-3 + εt� vt = -.0152*vt-1 + .1256*vt-2 + .3915*vt-3 + εt� MSE = 15.69082 (estimate of residual variance)

�� This estimate is smaller than the OLS estimate of 17.5656.

� The total R-square statistic computed from the residuals of the autoregressive model is 0.6256, reflecting the improved fit from the use of past residuals to help predict the next value of Qshrimpt.

� The Reg Rsq value is 0.5711, which is the R-square statistic for a regression of transformed variables adjusted for the estimated autocorrelation.61

Comparison of Diagnostic Statistics and Parameter Estimates from the Qshrimp Problem

Explanatory Variables OLS Yule-Walker Maximum Likelihood

Parameter Estimate

Standard Error

Parameter Estimate

Standard Error

Parameter Estimate

Standard Error

62 continued...

Intercept 29.6994 8.1180 29.6938 7.2391 30.4400 7.2224

Pshrimp -0.0322 0.003456 -0.0324 0.003706 -0.0341 0.004134

Pfin -0.0273 0.0136 -0.0177 0.0130 -0.0138 0.0135

Pshell1 0.0159 0.007185 0.009815 0.006848 0.007794 0.007022

Adshrimp -0.001014 0.007085 0.000814 0.006389 0.001561 0.006500

Adfin -0.0140 0.004805 -0.0151 0.004708 -0.0155 0.004768

Adshell1 0.008479 0.0208 0.001010 0.0193 -0.003312 0.0196

Comparison of Diagnostic Statistics and

Parameter EstimatesOLS Yule-Walker Maximum Likelihood

Parameter Estimate

Standard Error

Parameter Estimate

Standard Error

Parameter Estimate

Standard Error

MSE 17.5656 15.82434 15.69082

Root MSE 4.19113 3.97798 3.96116

SBC 578.028031 578.680834 578.057305

AIC 560.005054 552.933724 552.310196

Regress R-Square

0.5664 0.5618 0.5711

Total R-Square 0.5664 0.6224 0.6256

Log Likelihood -273.00253 -551.8663 -266.151

AR 1 NA 0.07152 0.101517 0.0152 0.1060

AR 2 NA -0.096118 0.101283 -0.1256 0.1037

AR 3 NA -0.321561 0.101517 -0.3915 0.1042

DW order 1 2.0921 2.0340 2.1042

DW order 2 1.8221 2.0047 2.0571

DW order 3 1.3585 1.8530 1.976263

Section 6.5 Section 6.5

Sample Problem:

The Demand for Gasoline

Model: MODEL1





Sum of Mean

Source DF Squares Square F Value Pr > F Source DF Squares Square F Value Pr > F

Model 6 0.78035 0.13006 150.85

Parameter Estimates

Parameter Standard


Intercept 1 -17.27084 1.71977 -10.04

Model: MODEL1



Pr < DW DW 1.0000






67

What is the conclusion regarding serial correlationbased on the Durbin-Watson statistic?

.02

.04

.06

RESID

-.06

-.04

-.02

.00

1960 1965 1970 1975 1980 1985 1990 199568

Name of Variable = lnpcgresName of Variable = lnpcgresName of Variable = lnpcgresName of Variable = lnpcgres

Mean of Working Series 9.68EMean of Working Series 9.68EMean of Working Series 9.68EMean of Working Series 9.68E----16161616

Standard Deviation 0.026354 Standard Deviation 0.026354 Standard Deviation 0.026354 Standard Deviation 0.026354

Number of Observations 36 Number of Observations 36 Number of Observations 36 Number of Observations 36

Autocorrelations Autocorrelations Autocorrelations Autocorrelations

Lag Covariance Correlation Lag Covariance Correlation Lag Covariance Correlation Lag Covariance Correlation ----1 9 8 7 6 5 41 9 8 7 6 5 41 9 8 7 6 5 41 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 3 2 1 0 1 2 3 4 5 6 7 8 9 1 3 2 1 0 1 2 3 4 5 6 7 8 9 1 3 2 1 0 1 2 3 4 5 6 7 8 9 1 Std ErrorStd ErrorStd ErrorStd Error

0 0.00069453 0 0.00069453 0 0.00069453 0 0.00069453 1.00000 1.00000 1.00000 1.00000 | |******** | |******** | |******** | |********************| ************| ************| ************| 0 0 0 0

1 1 1 1 0.00041753 0.00041753 0.00041753 0.00041753 0.60110.60110.60110.60117 7 7 7 | . | . | . | . |************ | |************ | |************ | |************ | 0.1666670.1666670.1666670.166667

2 0.00004865 2 0.00004865 2 0.00004865 2 0.00004865 0.07004 0.07004 0.07004 0.07004 | . | . | . | . |* . | |* . | |* . | |* . | 0.218760 0.218760 0.218760 0.218760

MA(1)

2 0.00004865 2 0.00004865 2 0.00004865 2 0.00004865 0.07004 0.07004 0.07004 0.07004 | . | . | . | . |* . | |* . | |* . | |* . | 0.218760 0.218760 0.218760 0.218760

3 3 3 3 ----0.0001150 0.0001150 0.0001150 0.0001150 ----.16552 .16552 .16552 .16552 | . | . | . | . ***| . | ***| . | ***| . | ***| . | 0.219382 0.219382 0.219382 0.219382

4 4 4 4 ----0.0001336 0.0001336 0.0001336 0.0001336 ----.19242 .19242 .19242 .19242 | . | . | . | . ****| . | ****| . | ****| . | ****| . | 0.222824 0.222824 0.222824 0.222824

5 5 5 5 ----0.0000673 0.0000673 0.0000673 0.0000673 ----.09685 .09685 .09685 .09685 | . | . | . | . **| . | **| . | **| . | **| . | 0.227393 0.227393 0.227393 0.227393

6 0.00004871 6 0.00004871 6 0.00004871 6 0.00004871 0.07014 0.07014 0.07014 0.07014 | . | . | . | . |* . | |* . | |* . | |* . | 0.2285360.2285360.2285360.228536

7 7 7 7 ----8.1403E8.1403E8.1403E8.1403E----7 7 7 7 ----.00117 .00117 .00117 .00117 | . | . | . | . | . | | . | | . | | . | 0.229133 0.229133 0.229133 0.229133

8 8 8 8 ----0.0001372 0.0001372 0.0001372 0.0001372 ----.19760 .19760 .19760 .19760 | . ****| | . ****| | . ****| | . ****| . | . | . | . | 0.229133 0.229133 0.229133 0.229133

9 9 9 9 ----0.0002564 0.0002564 0.0002564 0.0002564 ----.36917 .36917 .36917 .36917 | . ***| . ***| . ***| . *******| . | ****| . | ****| . | ****| . | 0.2338180.2338180.2338180.233818

69

Statistically significant autocorrelation and parti al autocorrelation coefficients

Partial Autocorrelations

Lag Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1

1 0.60117 | . |************ |

2 -0.45626 | *********| . |

3 0.09383 | . |** . |

4 -0.11796 | . **| . |

AR(1), AR(2)

4 -0.11796 | . **| . |

5 0.07910 | . |** . |

6 0.10048 | . |** . |

7 -0.33312 | *******| . |

8 -0.02557 | . *| . |

9 -0.31678 | .******| . |

70 continued...

Autocorrelation Check for White Noise

To Chi- Pr >

Lag Square DF ChiSq -----------------Autocorrelations------------

6 17.68 6 0.0071 0.601 0.070 -0.166 -0.192 -0.097 0.070

Statistically significant Ljung-Box Q* statistic.Thus, the pattern in the residuals is not random or white noise.

71


Dependent Variable lnpcg

Ordinary Least Squares Estimates

SSE 0.02500325 DFE 29

MSE 0.0008622 Root MSE 0.02936

SBC -134.55346 AIC -145.6381


Normal Test 0.5997 Pr > ChiSq 0.7409

Log Likelihood 79.8190475 Observations 36 Log Likelihood 79.8190475 Observations 36



1 0.7859



AR(1) 15.0740 0.0001

AR(2) 18.5985 |t|

Intercept 1 -17.2708 1.7198 -10.04


Estimates of Autocorrelations

Lag Covariance Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1

0 0.000695 1.000000 | |********************|

1 0.000418 0.601169 | |************ |

2 0.000049 0.070040 | |* |

Partial

Autocorrelations Autocorrelations

1 0.601169

2 -0.456258



Standard


1 -0.875457 0.171251 -5.11

2 0.456258 0.171251 2.66

75

Estimates of φφφφ1, φφφφ2

continued...

Yule-Walker Estimates

SSE 0.01185535 DFE 27

MSE 0.0004391 Root MSE 0.02095

SBC -153.33526 AIC -167.58693


Log Likelihood -27.396156 Observations 36

After you take



1 1.7378 0.0773 0.9227

2 2.2225 0.5771 0.4229

NOTE: PrDW

is the p-value for testing negative autocorrelation.

into account this pattern in the residuals, the serial correlation problem no longer is evident.

76



AR(1) 1.0852 0.2975

AR(2) 2.3248 0.3127

Standard Approx

The AUTOREG Procedure Output

Standard Approx


Intercept 1 -15.7683 1.8017 -8.75

Estimates of Autocorrelations

Lag Covariance Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1

0 0.000695 1.000000 | |********************|

1 0.000418 0.601169 | |************ |

2 0.000049 0.070040 | |* |

Partial

Autocorrelations

1 0.601169 Starting values in the 1 0.601169 2 -0.456258



Standard


1 -0.875457 0.171251 -5.11

2 0.456258 0.171251 2.66

78

Starting values in theML procedure to obtain estimates of φφφφ1, φφφφ2

continued...


SSE 0.01174976 DFE 27

MSE 0.0004352 Root MSE 0.02086

SBC -153.49238 AIC -167.74405


Log Likelihood 92.8720272 Observations 36

Durbin-Watson Statistics Durbin-Watson Statistics


1 1.8195 0.1255 0.8745

2 2.2171 0.5664 0.4336


79



AR(1) 0.7457 0.3878

AR(2) 2.0349 0.3615

Standard Approx


Intercept 1 -15.8652 1.9232 -8.25


Standard Approx


Intercept 1 -15.8652 1.7796 -8.92

Depiction of the Estimated Model for the Demand for Gasoline Problem (Greene)� The estimated model is based on the Maximum

Likelihood estimates:

tpptpuc

pncpgypcg

tt

tttt

*0109.0ln*0808.0ln*2007.0

ln*1494.0ln*0816.0ln*7770.18652.15ln

−−+−−+−=

vvv ε+−= *5194.0*9159.0

� Regress R-Square = 0.9580

� Total R-Square = 0.9854

MSE = 0.0004352 =

� DW order 1 = 1.8195

� DW order 2 = 2.2171

82

2σ̂

tttt vvv ε+−= −− 21 *5194.0*9159.0


A Test for Serial Correlation

in the Presence of a Lagged

Dependent Variable

Durbin’s h-TestA large sample test for autocorrelation when lagged dependent variables are present.

d is the DW statistic.))(nV1/(nˆh

d)2/1(1ˆ

β−ρ=−≈ρ

AR(1) process in error termsH0:ρ = 0

Coefficient associated with

The test breaks down if .

If the Durbin's h-test breaks down, compute the OLS residuals . Then regress on and the set of exogenous variables. The test

for ρ = 0 is carried out by testing the significance of the coefficient . (Durbin's m-test)

84

1tY −)1,0(N~h &

1)ˆ(nV ≥βtû

tû ,y,û 1t1t −−1tû −

Model: MODEL1




Number of Observations with Missing Values 1


Sum of Mean


Sum of Mean


Model 7 0.68226 0.09747 210.45

Parameter Estimates

Parameter Standard


Intercept 1 -6.81147 2.38667 -2.85 0.0082

laglnpcg 1 0.63554 0.12456 5.10

Model: MODEL1



Pr < DW 0.0079

Pr > DW 0.9921






87

The DW statistic reveals positive serial correlation, AR(1) pattern.

Name of Variable = lnpcgres

Mean of Working Series -66E-17

Standard Deviation 0.018902


Autocorrelations

Lag Covariance Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 Std Error

0 0.00035727 1.00000 | |********************| 0 0 0.00035727 1.00000 | |********************| 0

1 0.00006445 0.18040 | . |**** . | 0.169031

2 -0.0001012 -.28328 | .******| . | 0.174445

3 -0.0000410 -.11466 | . **| . | 0.187127

4 -0.0000612 -.17122 | . ***| . | 0.189124

5 0.00002013 0.05634 | . |* . | 0.193502

6 0.00007480 0.20936 | . |**** . | 0.193970

7 0.00002450 0.06858 | . |* . | 0.200323

8 -0.0000229 -.06416 | . *| . | 0.200992

88

Partial Autocorrelations

Lag Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1

1 0.18040 | . |**** . |

2 -0.32645 | *******| . |

3 0.01390 | . | . |

4 -0.27677 | .******| . |

5 0.15471 | . |*** . |

6 0.02360 | . | . |

7 0.07670 | . |** . |

From the ACF and PACF plots, no pattern in the resi duals is revealed.

7 0.07670 | . |** . |

8 -0.06233 | . *| . |

The ARIMA Procedure

Autocorrelation Check for White Noise

To Chi- Pr >

Lag Square DF ChiSq -----------------Autocorrelations-------------

6 8.24 6 0.2211 0.180 -0.283 -0.115 -0.171 0.056 0.209

The Ljung-Box Q* statistic suggests a white noise r esidual series.89


Dependent Variable lnpcg

Ordinary Least Squares Estimates

SSE 0.01250434 DFE 27

MSE 0.0004631 Root MSE 0.02152

SBC -150.02748 AIC -162.47027

Durbin’s h-test, at least at the 0.10 level of significance, suggests a non-random residual series.

SBC -150.02748 AIC -162.47027


Durbin h 1.5788 Pr > h 0.0572

Normal Test 1.3963 Pr > ChiSq 0.4975


Durbin-Watson 1.6391



AR(1) 1.7213 0.1895 90 continued...

Standard Approx


Intercept 1 -6.8115 2.3867 -2.85 0.0082

laglnpcg 1 0.6355 0.1246 5.10

Partial

Autocorrelations

1 0.180402



Starting value of φφφφin the ML procedure


Standard


1 -0.180402 0.192898 -0.94

Algorithm converged. 92


SSE 0.00983118 DFE 26

MSE 0.0003781 Root MSE 0.01945

SBC -153.04594 AIC -167.04408



Durbin-Watson 1.7302



AR(1) 2.3826 0.1227

93

GLS (ML) Estimates

continued...

Standard Approx


Intercept 1 -8.3477 2.5227 -3.31 0.0027

laglnpcg 1 0.2081 0.1299 1.60 0.1213

lny 1 0.9295 0.2890 3.22 0.0035

lnpg 1 -0.2157 0.0407 -5.31



Standard Approx


Intercept 1 -8.3477 2.2511 -3.71 0.0010

laglnpcg 1 0.2081 0.1223 1.70 0.1008

95

laglnpcg 1 0.2081 0.1223 1.70 0.1008

lny 1 0.9295 0.2567 3.62 0.0012

lnpg 1 -0.2157 0.0373 -5.79

Calculation of Durbin’s h-Test for the Greene Problem

In the Greene Problem for gasoline demand:

35

1805.0)2/639.1(1ˆ

=

=−=

n

ρ

96

5788.1

0155.0)12456.0()(

35

2

=

==

=

h

Bv

n


Sum of Mean


Model 7 0.00061193 0.00008742 0.19 0.9848

Error 26 0.01189 0.00045736

Corrected Total 33 0.01250

Root MSE 0.02139 R-Square 0.0489

Dependent Mean -0.00002950 Adj R-Sq -0.2071

Coeff Var -72488

Parameter Estimates

Parameter Standard

Variable Label DF Estimate Error t Value Pr > |t|

Intercept Intercept 1 -1.10845 2.12226 -0.52 0.6059

lnpcgres1 1 0.26893 0.23259 1.16 0.2581

laglnpcg 1 -0.08946 0.13979 -0.64 0.5278

lny 1 0.12258 0.23732 0.52 0.6098

lnpg 1 0.00625 0.02344 0.27 0.7918

lnpnc 1 -0.02989 0.05719 -0.52 0.6056

lnppt 1 -0.00223 0.06428 -0.03 0.9726

t 1 0.00038584 0.00450 0.09 0.9324 97


Summary Remarks about the

Issue of Serial Correlation

Final Considerations� With time-series data, in most cases this problem will surface.

� Analysts must examine the error structure carefully.

� Minimally do the following:

– Graph the residuals over time.

– Consider the significance of the Durbin-Watson statistic.

– Consider higher-order autocorrelation structure via PROC ARIMA.ARIMA.

– Consider the Godfrey LM test.

– Consider the Box-Pierce or Ljung-Box tests (Q statistics).

� Re-estimate econometric models with AR(p) error structures via PROC AUTOREG.

– Use the Yule-Walker or Maximum Likelihood method to obtain estimates of the AR(P) error structure.

– A preference exists for the Maximum Likelihood method.

99

becin 002 chapter 6 · used test for serial correlation, there are limitations: 1. the test is for...

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