AUTOCORRELATION · PDF file · 2017-10-25Consequences of autocorrelation Detecting autocorrelation Remedial measures ... This method still uses OLS, but corrects the standard errors

AUTOCORRELATION

Phung Thanh Binh

▪ Time series Gauss-Markov conditions

▪ The nature of autocorrelation

▪ Causes of autocorrelation

▪ Consequences of autocorrelation

▪ Detecting autocorrelation

▪ Remedial measures

OUTLINE

3) No perfect collinearity

2) Zero conditional mean of error: E[ut|Xjk] = 0

1) Linear: Yt = 1 + 2X2t + 3X3t + ut

Unbiased

5) No serial correlation: Cov(ut,us|Xjk) = 0

4) Homoscedasticity: Var(ut|Xjk) = 2

if violated => ARCH family models: Lecture 16 (for T.S data)

BLUE

Lecture 13violated

Time series Gauss Markov conditions

▪ Terminological issue:

▪ Common practice: the terms autocorrelation and

serial correlation are the same.

▪ Some authors prefer to distinguish the two terms as

follows:

▪ Autocorrelation: Lag correlation of a given series with

itself, lagged by a number of time units.

▪ Serial correlation: Lag correlation between two different

series.

The nature of autocorrelation




Positive

autocorrelation

Negative

autocorrelation

No

autocorrelation

The meaning of ρ: The error term ut at time t is a linear

combination of the current and past disturbance.

▪ The possible strong correlation between the

observation i with the observation j could be due to:

▪ Inertia

▪ Cobweb phenomenon

▪ Lags

▪ Nonstationarity

▪ Specification bias: Excluded variables case

▪ Specification bias: Incorrect functional form

▪ etc … [see Gujarati (2009). Basic Econometrics]

Pure autocorrelation

Impure

autocorrelation

The causes of autocorrelation

The consequences of autocorrelation

▪ The estimated coefficients are still unbiased.

▪ The variance of the is no longer the smallest.

▪ The standard error of the estimated coefficient,

becomes large.

The consequences of autocorrelation

Table6_1.dta (in Econometrics by examples)

Detecting autocorrelation

▪ Graphical method

▪ Plot the values of the residuals, et, chronologically

▪ If discernible pattern exists, autocorrelation likely a

problem.

▪ Durbin-Watson test: Durbin-Watson’s d statistic

▪ Durbin’s h statistic

▪ Breusch-Godfrey (BG) test

-2-1

01

23

0 20 40 60Time

Residuals Standardized residuals

. predict s1, resid

. gen s1_100=100*s1

. label var s1_100 "Residuals"

. predict s2, rstandard

. twoway (line s1_100 time) (line s2 time)

Assumptions are:

1. The regression model includes an intercept term.

2. The regressors are fixed in repeated sampling.

3. The error term follows the first-order autoregressive

(AR1) scheme:

4. The regressors do not include the lagged value(s) of

the dependent variable, Yt.

5. No missing observation.

1t t tu u v

Durbin-Watson d Statistic


d (ei ei1)

2ei

2, for n and K -1 d.f.

Positive Zone of No Autocorrelation Zone of Negative

autocorrelation indecision indecision autocorrelation

|_______________|__________________|_____________|_____________|__________________|___________________|

0 d-lower d-upper 2 4-d-upper 4-d-lower 4

Autocorrelation is clearly evident

Ambiguous – cannot rule out autocorrelation

Autocorrelation in not evident


▪ This test allows for:

(1) Lagged values of the dependent variables to be included as

regressors

(2) Higher-order autoregressive schemes, such as AR(2), AR(3),

etc.

(3) Moving average terms of the error term, such as ut-1, ut-2, etc.

▪ The error term in the main equation follows the following AR(p)

autoregressive structure:

▪ The null hypothesis of no serial correlation is:

1 2 ... 0p

1 1 2 2 ...t t t p t p tu u u u v

Breusch-Godfrey (BG) test

The BG test involves the following steps:

▪ Regress et, the residuals from our main regression, on the

regressors in the model and the p autoregressive terms given in

the equation on the previous slide, and obtain R2 from this

auxiliary regression.

▪ If the sample size is large, BG have shown that: (n – p)R2 ~ X2p

▪ That is, in large samples, (n – p) times R2 follows the chi-square distribution with

p degrees of freedom.

▪ Rejection of the null hypothesis implies evidence of

autocorrelation.

▪ As an alternative, we can use the F value obtained from the

auxiliary regression.

▪ This F value has (p , n-k-p) degrees of freedom in the numerator and

denominator, respectively, where k represents the number of parameters in the

auxiliary regression (including the intercept term).



▪ First-Difference Transformation

▪ If autocorrelation is of AR(1) type, we have:

▪ Assume ρ=1 and run first-difference model (taking first difference

of dependent variable and all regressors)

▪ Generalized Transformation

▪ Estimate value of ρ through regression of residual on lagged

residual and use value to run transformed regression

▪ Newey-West Method

▪ Generates HAC (heteroscedasticity and autocorrelation

consistent) standard errors

▪ Model Evaluation

1t t tu u v

Remedial Measures

First-Difference Method

▪ This outcome could be due to the wrong value of ρ (ρ

= 1) chosen for transformation.

▪ Notes:

▪ There if no intercept in the first-difference model.

▪ If there is an intercept term, what does it stand for?

▪ Rule of thumb:

▪ Use the first-difference form whenever d < R2 (Maddala).

▪ Use the first-difference form when ut is nonstationary (or

differently, Yt and Xt are not cointegrated).

First-Difference Method

Feasible Generalized Least Squares (FGLS)

Feasible Generalized Least Squares (FGLS)

FGLS: Prais-Winsten Transformation

FGLS: Cochrane-Orcutt Transformation

One of the iterative

methods of

estimating ρ

FGLS: Cochrane-Orcutt Transformation

How does the Cochrane-Orcutt procedure work?


Prais-Winsten

procedure is

similar, but it

transforms the

first observation

differently.


The Newey-West Method

▪ This method still uses OLS, but corrects the

standard errors for autocorrelation.

▪ This is an extension of White’s heteroscedasticity-

consistent standard errors.

▪ The corrected standard errors are known as HAC

(heteroscedasticity- and autocorrelation-consistent)

standard errors or simply Newey-West standard

errors.

▪ This is strictly valid in large samples.



AUTOCORRELATION · PDF file · 2017-10-25Consequences of autocorrelation Detecting autocorrelation Remedial measures ... This method still uses OLS, but corrects the standard errors

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