chapter 9

ECON 6002Econometrics Memorial University of Newfoundland

Adapted from Vera Tabakova’s notes

9.1 Introduction

9.2 Lags in the Error Term: Autocorrelation

9.3 Estimating an AR(1) Error Model

9.4 Testing for Autocorrelation

9.5 An Introduction to Forecasting: Autoregressive Models

9.6 Finite Distributed Lags

9.7 Autoregressive Distributed Lag Models

Figure 9.1

The model so far assumes that the observations are not correlated with one another.

This is believable if one has drawn a random sample, but less likely if one has drawn observations sequentially in time

Time series observations, which are drawn at regular intervals, usually embody a structure where time is an important component.

Principles of Econometrics, 3rd Edition

If we cannot completely model this structure in the regression function itself, then the remainder spills over into the unobserved component of the statistical model (its error) and this causes the errors be correlated with one another.

Principles of Econometrics, 3rd Edition

1 2( , , ,...)t t t ty f x x x

1( , )t t ty f y x

1( ) ( )t t t t ty f x e e f e

Three ways to view the dynamics:

Figure 9.2(a) Time Series of a Stationary Variable

Assume Stationarity For now

Figure 9.2(b) Time Series of a Nonstationary Variable that is ‘Slow Turning’ or ‘Wandering’

Figure 9.2(c) Time Series of a Nonstationary Variable that ‘Trends’

9.2.1 Area Response Model for Sugar Cane

1 2ln lnA P

1 2ln lnt t tA P e

1 2t t ty x e

1t t te e v

1 2t t ty x e

1t t te e v

2( ) 0 var( ) cov( , ) 0 fort t v t sE v v v v t s

1 1 Assume also

( ) 0tE e

22

2var( )

1v

t ee

2cov , 0kt t k ee e k

OK

OK, constant

Not zero!!!

2

2

cov( , ) cov( , )corr( , )

var( )var( ) var

kkt t k t t k e

t t kt et t k

e e e ee e

ee e

1corr( , )t te e

ˆ 3.893 .776

(se) (.061) (.277)t ty x

Aka autocorrelation coefficient

Figure 9.3 Least Squares Residuals Plotted Against Time

1

2 2

1 1

( )( )cov( , )

var( )var( ) ( ) ( )

T

t tt t t

xy T T

t tt t

t t

x x y yx y

rx y x x y y

11 2

12

12

ˆ ˆcov( , )

var( ) ˆ

T

t tt t t

T

t tt

e ee e

re e

Constant varianceNull expectation of x and y

The existence of AR(1) errors implies:

The least squares estimator is still a linear and unbiased estimator, but

it is no longer best. There is another estimator with a smaller

variance.

The standard errors usually computed for the least squares estimator

are incorrect. Confidence intervals and hypothesis tests that use these

standard errors may be misleading.

As with heteroskedastic errors, you can salvage OLS when your data are autocorrelated.

Now you can use an estimator of standard errors that is robust to both heteroskedasticity and autocorrelation proposed by Newey and West.

This estimator is sometimes called HAC, which stands for heteroskedasticity autocorrelated consistent.

HAC is not as automatic as the heteroskedasticity consistent (HC) estimator.

With autocorrelation you have to specify how far away in time the autocorrelation is likely to be significant

The autocorrelated errors over the chosen time window are averaged in the computation of the HAC standard errors; you have to specify how many periods over which to average and how much weight to assign each residual in that average.

That weighted average is called a kernel and the number of errors to average is called bandwidth.

Usually, you choose a method of averaging (Bartlett kernel or Parzen kernel) and a bandwidth (nw1, nw2 or some integer). Often your software defaults to the Bartlett kernel and a bandwidth computed based on the sample size, N.

Trade-off: Larger bandwidths reduce bias (good) as well as precision (bad). Smaller bandwidths exclude more relevant autocorrelations (and hence have more bias), but use more observations to increase precision (smaller variance).

Choose a bandwidth large enough to contain the largest autocorrelations. The choice will ultimately depend on the frequency of observation and the length of time it takes for your system to adjust to shocks.

Sugar cane example

The two sets of standard errors, along with the estimated equation are:

The 95% confidence intervals for β2 are:

ˆ 3.893 .776

(.061) (.277) 'incorrect' se's

(.062) (.378) 'correct' se's

t ty x

(.211,1.340) (incorrect)

(.006,1.546) (correct)

1 2t t ty x e

1t t te e v

1 2 1t t t ty x e v

1 1 1 2 1t t te y x

1 1 1 2 1t t te y x

1 2 1 2 1(1 )t t t t ty x y x v

1ln( ) 3.899 .888ln( ) .422

(se) (.092) (.259) (.166)t t t t tA P e e v

Now this transformed nonlinear model has errors that are uncorrelated over time

It can be shown that nonlinear least squares estimation of (9.24) is

equivalent to using an iterative generalized least squares estimator

called the Cochrane-Orcutt procedure. Details are provided in

Appendix 9A.

The nonlinear least squares estimator only requires that the errors be stable (not necessarily stationary).

Other methods commonly used make stronger demands on the data, namely that the errors be covariance stationary.

Furthermore, the nonlinear least squares estimator gives you an unconditional estimate of the autocorrelation parameter, , and yields a simple t-test of the hypothesis of no serial correlation.

Monte Carlo studies show that it performs well in small samples as well.

But nonlinear least squares requires more computational power than linear estimation, though this is not much of a constraint these days.

Nonlinear least squares (and other nonlinear estimators) use numerical methods rather than analytical ones to find the minimum of your sum of squared errors objective function. The routines that do this are iterative.

1 2 2 1 1(1 )t t t t ty x x y v

0 1 1 1 1t t t t ty x x y v

1 0 2 1 2 1(1 )

1 1ˆ 2.366 .777 .611 .404

(se) (.656) (.280) (.297) (.167)

t t t ty x x y We should then test the above restrictions

9.4.1 Residual Correlogram

0 1: 0 : 0H H

1 (0,1)z T r N

34 .404 2.36 1.96z

9.4.1 Residual Correlogram (more practically…)

1 1

1.96 1.96 or r r

T T

1.96 1.96 or k kr r

T T

2

cov( , ) ( )

var( ) ( )t t k t t k

kt t

e e E e e

e E e

Figure 9.4 Correlogram for Least Squares Residuals from Sugar Cane Example

1 2t t ty x e

1 2 1 2 1(1 )t t t t ty x y x v

For this nonlinear model, then, the residuals should be uncorrelated

Figure 9.5 Correlogram for Nonlinear Least Squares Residualsfrom Sugar Cane Example

The other way to determine whether or not your residuals are autocorrelated is to use an LM (Lagrange multiplier) test. For autocorrelation, this test is based on an auxiliary regression where lagged OLS residuals are added to the original regression equation. If the coefficient on the lagged residual is significant then you conclude that the model is autocorrelated.

1 2 1t t t ty x e v

= 2.439 = 5.949 -value = .021t F p

1 2 1ˆ ˆt t t ty x e v

1 2 1 2 1ˆ ˆ ˆt t t t tb b x e x e v

We can derive the auxiliary regression for the LM test:

1 1 2 2 1

1 2 1

ˆ ˆ ˆ( ) ( )

ˆ ˆ

t t t t

t t t

e b b x e v

x e v

2 34 .16101 5.474LM T R

Centered around zero, so the power of this test now would come from The lagged error, which is what we care about…

1 1 2 2 1

1 2 1

ˆ ˆ ˆ( ) ( )

ˆ ˆ

t t t t

t t t

e b b x e v

x e v

2 34 .16101 5.474LM T R This was the Breusch-Godfrey LM test for autocorrelation and it is distributed chi-sq

In STATA:regress la lpestat bgodfrey

1 1 2 2t t t p t p ty y y y v

11

1

ln( ) ln( ) 100 100t tt t t

t

CPI CPIy CPI CPI

CPI

1 2 3.1883 .3733 .2179 .1013

(se) (.0253) (.0615) (.0645) (.0613)

t t t tINFLN INFLN INFLN INFLN

Adding lagged values of y can serve to eliminate the error autocorrelation

In general, p should be large enough so that vt is white noise

Figure 9.6 Correlogram for Least Squares Residuals fromAR(3) Model for Inflation

Helps decide How many lags To include in the model

1 1 2 2 3 3t t t t ty y y y v

1 1 2 1 3 2 1T T T T Ty y y y v

1 1 2 1 3 2ˆ ˆ ˆ ˆˆ

.1883 .3733 .4468 .2179 .5988 .1013 .3510

.2602

T T T Ty y y y

2 1 1 2 3 1ˆ ˆ ˆ ˆˆ ˆ

.1883 .3733 .2602 .2179 .4468 .1013 .5988

.2487

T T T Ty y y y

1 1 1 1 1 2 2 1 3 3 2 1ˆ ˆ ˆ ˆˆ ( ) ( ) ( ) ( )T T T T T Tu y y y y y v

1 1Tu v

2 1 1 1 2 1 1 2 1 1 2ˆ( )T T T T T Tu y y v u v v v

23 1 2 2 1 3 1 2 1 1 2 3( )T T T Tu u u v v v v

2 21 1

2 2 22 2 1

2 2 2 2 23 3 1 2 1

var( )

var( ) (1 )

var( ) [( ) 1]

v

v

v

u

u

u

ˆ ˆ ˆ ˆ1.96 , 1.96T j j T j jy y

This model is just a generalization of the ones previously discussed.

In this model you include lags of the dependent variable (autoregressive) and the contemporaneous and lagged values of independent variables as regressors (distributed lags).

The acronym is ARDL(p,q) where p is the maximum distributed lag and q is the maximum autoregressive lag

0 1 1 2 2 , 1, ,t t t t q t q ty x x x x v t q T

( )ts

t s

E y

x

11

1

ln( ) ln( ) 100 100t tt t t

t

WAGE WAGEx WAGE WAGE

WAGE

0 1 1 1 1t t t q t q t p t p ty x x x y y v

0 1 1 2 2 3 3

0

t t t t t t

s t s ts

y x x x x e

x e

Figure 9.7 Correlogram for Least Squares Residuals fromFinite Distributed Lag Model

1 2

3 1 2

.0989 .1149 .0377 .0593

(se) (.0288) (.0761) (.0812) (.0812)

.2361 .3536 .1976

(.0829) (.0604) (.0604)

t t t t

t t t

INFLN PCWAGE PCWAGE PCWAGE

PCWAGE INFLN INFLN

Figure 9.8 Correlogram for Least Squares Residuals from Autoregressive Distributed Lag Model

0 1 1 2 2 3 3 1 1 2 2t t t t t t t ty x x x x y y v

0 0

1 1 0 1

2 1 1 2 0 2

3 1 2 2 1 3

4 1 3 2 2

ˆ ˆ .1149

ˆ ˆ ˆ ˆ .3536 .1149 .0377 .0784

ˆ ˆ ˆ ˆ ˆ ˆ .0643

ˆ ˆ ˆ ˆ ˆ ˆ .2434

ˆ ˆ ˆ ˆ ˆ .0734

Figure 9.9 Distributed Lag Weights for Autoregressive Distributed Lag Model

54Principles of Econometrics, 3rd Edition


(9A.2)

1 2 1 t t t t t ty x e e e v

(9A.1)1 2 1 1 2 1t t t t ty x y x v

1 1 2 11t t t t ty y x x v

1 2 1 1 1t t t t t t ty y y x x x x


(9A.4)

(9A.3)1 1 2 2t t t ty x x v

1 2 1 1 2 1( )t t t t ty x y x v


(9A.5)

1 1 1 2 1y x e

2 2 2 21 1 1 2 11 1 1 1y x e

1 11 1 12 2 1y x x e

(9A.6)

2 21 1 11

2 212 1 1 1

1 1

1 1

y y x

x x e e


22 2 2

1 1 2var( ) (1 ) var( ) (1 )

1v

ve e


(9B.1)

0 1: 0 : 0H H

2

12

2

1

ˆ ˆ

ˆ

T

t tt

T

tt

e ed

e


(9B.2)

2 21 1

2 2 2

2

1

2 21 1

2 2 2

2 2 2

1 1 1

1

ˆ ˆ ˆ ˆ2

ˆ

ˆ ˆ ˆ ˆ2

ˆ ˆ ˆ

1 1 2

T T T

t t t tt t t

T

tt

T T T

t t t tt t tT T T

t t tt t t

e e e ed

e

e e e e

e e e

r


(9B.3) 12 1d r

cd d

Figure 9A.1:

Principles of Econometrics, 3rd Edition Slide 9-63

Figure 9A.2:


The Durbin-Watson bounds test.

if the test is inconclusive.


0 1if , reject : 0 and accept : 0;Lcd d H H

0if , do not reject : 0;Ucd d H

,Lc Ucd d d


0 1 1 2 2 3 30

t t t t t t s t s ts

y x x x x e x e

0 1 1 1 1t t t q t q t p t p ty x x x y y v


(9C.2)

(9C.1)0 1 1t t t ty x y v

1 0 1 1 2t t ty x y

0 1 1 0 1 0 1 1 2

21 0 1 0 1 1 2

( )t t t t t t

t t t

y x y x x y

x x y


(9C.3)

21 0 1 0 1 1 0 2 1 3

2 2 31 1 0 1 0 1 1 0 2 1 3

( )t t t t t

t t t t

y x x x y

x x x y

21 1 1

2 10 1 0 1 1 0 2 1 0 1 ( 1)

2 11 1 1 0 1 1 ( 1)

0

(1 )

jt

j jt t t t j t j

jj s j

t s t js

y

x x x x y

x y


(9C.4)0 10

st t s

s

y x

21 1

1

(1 )1

0t s t s t

s

y x e


0 1s

s

2 00 1 1

0 1

(1 )1s

s


(9C.6)

(9C.5)0 1 1 2 2 3 3 1 1 2 2t t t t t t t ty x x x x y y v

0 0

1 1 0 1

2 1 1 2 0 2

3 1 2 2 1 3

4 1 3 2 2

1 1 2 2 for 4s s s s


(9D.2)

(9D.1)

1 21ˆ

3T T T

T

y y yy

1 21 1 2ˆ (1 ) (1 )T T T Ty y y y

2 31 2 3ˆ(1 ) (1 ) (1 ) (1 ) .....T T T Ty y y y

1ˆ ˆ(1 )T T Ty y y

Figure 9A.3: Exponential Smoothing Forecasts for two alternative values of α


chapter 9

Documents

yprinciples of econometrics

nowprinciples of econometrics

timeprinciples of econometrics

alsoprinciples of econometrics

estimator of standard

squares estimator

hac standard errors

heteroskedastic errors