Autocorrelation in Linear Regression: Identification, Consequences, and Detection, Summaries of Economics

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REGRESSION DIAGNOSTIC III:

AUTOCORRELATION

AUTOCORRELATION

 (^) One of the assumptions of the classical linear regression (CLRM) is that the covariance between u i , the error term for observation i, and u j , the error term for observation j, is zero.  (^) Reasons for autocorrelation include:  (^) The possible strong correlation between the shock in time t with the shock in time t+  (^) More common in time series data

DETECTION OF AUTOCORRELATION

There are two ways in general. The first is the informal way which is done through graphs and therefore we call it the graphical method or Visual Inspection. ACF/PACF PLOTS The second is through formal tests for autocorrelation, like the following ones:

1. The Durbin Watson Test
2. The Breusch-Godfrey Test
3. Q-stats

DURBIN-WATSON ( d ) TEST

 (^) The Durbin-Watson d statistic is defined as:

First-Order Autocorrelation (AR-1 PROCESS)

The simplest and most commonly observed is the first-order autocorrelation. Consider the multiple regression model: Y t =β 1 +β 2

X

2t +β 3

X

3t +β 4

X

4t +…+β k

X

kt +u t in which the current observation of the error term u t is a function of the previous (lagged) observation of the error term: u t =ρu t- +e t

First-Order Autocorrelation

The coefficient ρ is called the first-order autocorrelation coefficient and takes values from -1 to +1. It is obvious that the size of ρ will determine the strength of serial correlation. We can have three different cases. (a)If ρ is zero, then we have no autocorrelation. (b) If ρ approaches unity, the value of the previous observation of the error becomes more important in determining the value of the current error and therefore high degree of autocorrelation exists. In this case we have positive autocorrelation. (c) If ρ approaches -1, we have high degree of negative autocorrelation.

DW DECISION RULES

DW DECISION RULES

 (^) If the statistic lies near the value 2, there is no serial correlation.  (^) But if the statistic lies in the vicinity of 0, there is positive serial correlation.  (^) The closer the d is to zero, the greater the evidence of positive serial correlation.  (^) If it lies in the vicinity of 4, there is evidence of negative serial correlation

 Role of thumb the usual range of DW test statistics is

The Breusch-Godfrey Test

It is a Lagrange Multiplier Test that resolves the drawbacks of the DW test. Consider the model: Y t =β 1 +β 2

X

2t +β 3

X

3t +β 4

X

4t +…+β k

X

kt +u t where: u t =ρ 1 u t-

• ρ 2 u t- +ρ 3 u t- +…+ ρ p u t-p +e t

Combining those two we get:

Y

t

1

2

X

2t

3

X

3t

4

X

4t

k

X

kt

1

u

t-

2

u

t-

3

u

t-

p

u

t-p

+e

t

The null and the alternative hypotheses are:

H

0

1

2

p

=0 no autocorrelation

H

a

: at least one of the ρ’s is not zero, thus, autocorrelation

The Breusch-Godfrey Test

Step 1: Estimate the model and obtain the residuals Step 2: Run the full LM model with the number of lags used being determined by the assumed order of autocorrelation. Step 3: Compute the LM statistic = (n-ρ)R 2 from the LM model and compare it with the chi-square critical value. Step 4: Conclude