Autocorrelation

Lecture 20 1

Econ 140Econ 140

Autocorrelation

Lecture 20

Lecture 20 2

Econ 140Econ 140Today’s plan

• Definition and implications

• How to test for first order autocorrelation

– Note: we’ll only be taking a detailed look at 1st order autocorrelation, but higher orders exist

– e.g. quarterly data is likely to have 4th order autocorrelation

• How to correct for first-order autocorrelation and how to estimate allowing for autocorrelation

• Again we’ll use the Phillips curve as an example

Lecture 20 3

Econ 140Econ 140Definitions and implications

• Autocorrelation is a time-series phenomenon

• 1st-order autocorrelation implies that neighboring observations are correlated

– the observations aren’t independent draws from the sample

Lecture 20 4

Econ 140Econ 140Definitions and implications (2)

• In terms of the Gauss-Markov (or BLUE) theorem:

• The model is still linear and unbiased if autocorrelation exists:

2 whereˆ

t

tttt

x

xcYcb

tt bXaY :is model the

bbXacbE tt ˆ

Lecture 20 5


• Autocorrelation will affect the variance:

• if s = t, then we would have:

• but if s t, and Y observations are not independent, we have nonzero covariance terms:

s t sttsccbV ˆ

22tc

s t sttscc 2

Lecture 20 6


• Think of a numerical example to demonstrate this

• assuming t = 3: if we wanted to estimate :

• We want to consider the efficiency, or the variance of

– If BLUE: all covariance terms are zero.

– If covariance terms are nonzero: we no longer have minimum variance

– minimum variance is defined as

YtcYcYcYcb t332211ˆ

b̂

b̂

s t sttst cccbV 2ˆ 22

22tc

Lecture 20 7

Econ 140Econ 140Summary of implications

1) Estimates are linear and unbiased

2) Estimates are not efficient. We no longer have minimum variance

3) Estimated variances are biased either positively or negatively

4) Unreliable t and F test results

5) Computed variances and standard errors for predictions are biased

• Main idea: autocorrelation affects the efficiency of estimators

Lecture 20 8

Econ 140Econ 140How does autocorrelation occur?

Autocorrelation occurs through one of the following avenues:

1) Intertia in economic series through construction

– With regards to unemployment this was called hysteresis: this means that certain sections of society who are prone to unemployment

2) Incorrect model specification

– There might be missing variables or we might have transformed the model to create correlation across variables

Lecture 20 9

Econ 140Econ 140How does autocorrelation occur?

3) Cobweb phenomenon

– agents respond to information with lags to

– this is usually related to agricultural markets

4) Data manipulation

– example: constructing annual information based on quarterly data

Lecture 20 10

Econ 140Econ 140Graphical results

• With no autocorrelation in the error term: we would expect all errors to be randomly dispersed around zero within reasonable boundaries

• Simply graphing the estimated errors against time indicates the possibility of autocorrelation:

– we look for patterns of errors over time

– patterns can be positive, negative, or zero

• Graph error vs. time, we have positive correlation in the error term

– errors from one time period to the next tend to move in 1 direction, with a positive slope

Lecture 20 11

Econ 140Econ 140Phillips Curve

• L_20.xls : Phillips Curve data

– Can calculate predicted wage inflation using the observed unemployment rate and the estimated regression coefficients

– Can then calculate the estimated error of the regression equation

– Can also calculate the error lagged one time period

Lecture 20 12

Econ 140Econ 140Durbin-Watson statistic

• We will use the Durbin-Watson statistic to test for autocorrelation

• This is computed by looking over T-1 observations where t = 1, …T

Tt t

tt

e

eed

Tt

22

21

ˆ

ˆˆ2

Lecture 20 13

Econ 140Econ 140Durbin-Watson statistic (2)

• The assumptions behind the Durbin-Watson statistic are:

1) You must include an intercept in the regression

2) Values of X are independent and fixed

3) Disturbances, or errors, are generated by:

• this says that errors in this time period are correlated with errors in the last time period and some random error vt

is the coefficient of autocorrelation and is bounded -1 1

can be calculated as

vee tt 1

Tt t

Tt tt

e

ee

22

2 1

ˆ

ˆˆ̂

Lecture 20 14

Econ 140Econ 140How to estimate

• This estimation matters because it will be used in the model correction

can be estimated by this equation:

• Once we have the Durbin-Watson statistic d, you can obtain an estimate for

21ˆ d

Lecture 20 15

Econ 140Econ 140How to estimate (2)

• How the test works:

• The values for d range between 0 and 4 with 2 as the midpoint

– using : 4,2

1ˆ dd

value d value Autocorrelation?

-1 4 Perfect negative autocorrelation

0 2 Zero autocorrelation

1 0 Perfect positive autocorrelation

Lecture 20 16

Econ 140Econ 140How to estimate (3)

• We can represent this in the following figure:

2 40

H1

dL dU

H0: =0Reject null Cannot reject null

4-dU 4-dL

Reject nullH1

indeterminate indeterminate

• dL represents the D-W upper bound

• dU represents the D-W lower bound

• The mirror image of dL and dU are 4- dL and 4-dU

Lecture 20 17

Econ 140Econ 140Procedure

Table on the second handout for today is the Durbin-Watson statistical table and an additional table for this analysis

1) Run model:

2) Compute:

3) Compute d statistic

4) Find dL and dU from the tables K’ is the number of parameters minus the constant and T is the number of observations

5) Test to see if autocorrelation is present

ttt ebXaY

ttt YYe ˆˆ

Lecture 20 18

Econ 140Econ 140Example (2)

• Returning to L_20.xls

2 40

H1

dL dU

H0: =0Reject null Accept null

4-dU 4-dL

Reject nullH1

0.3311.475

47 1'

475.1 331.0

nsobservatioK

dd L

Lecture 20 19

Econ 140Econ 140Generalized least squares

• What can we do about autocorrelation?

• Recall that our model is: Yt = a + bXt + et (1)

• We also know: et = et-1 + vt

• We will have to estimate the model using generalized least squares (GLS)

Lecture 20 20

Econ 140Econ 140Generalized least squares (2)

• Let’s take our model and lag it by one time period:

Yt-1 = a + bXt-1 + et-1

• Multiplying by :

Yt-1 = a + bXt-1 + et-1 (2)

• Subtracting our (2) from (1), we get

Yt - Yt-1 = a(1-) + b(Xt-1 - Xt-1) + vt

where

vt = et - et-1

Lecture 20 21

Econ 140Econ 140Generalized least squares (3)

• Now we need an estimate of : we can transform the variables such that:

where:

• Estimating equation (3) allows us to estimate without first-order autocorrelation

(3) * ***ttt ebXaY

1*

ttt YYY

Lecture 20 22

Econ 140Econ 140Estimating

• There are several approaches

• One way is by using a short cut:

thinking back to the Durbin-Watson statistic,

we can rewrite the expression for d as:

21ˆ

d

Tt t

tt

e

eed

Tt

22

21

ˆ

ˆˆ2

Tt t

Tt t

Tt

Tt tt

Tt t

Tt t

e

e

e

ee

e

ed

22

22

1

22

2 1

22

22

ˆ

ˆ

ˆ

ˆˆ2

ˆ

ˆ

Lecture 20 23

Econ 140Econ 140Estimating (2)

• Collecting like terms, we have:

• Solving for , we can get an estimate in terms of d:

• Since earlier we defined as:

– we can use this to get a more precise estimate

• There are three or four other methods in the text

d ̂22

T

t

Tt tt

e

ee

22

2 1

ˆ

ˆˆ̂

21ˆ

d

Documents

Autocorrelation