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

The nature of autocorrelation

The nature of autocorrelation

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

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

Durbin-Watson d Statistic

▪ 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).

Breusch-Godfrey (BG) test

Breusch-Godfrey (BG) test

▪ 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?

How does the Cochrane-Orcutt procedure work?

Prais-Winsten

procedure is

similar, but it

transforms the

first observation

differently.

How does the Cochrane-Orcutt procedure work?

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.

The Newey-West Method

The Newey-West Method