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Hill, Griffiths & Lim, Principles of Econometrics, 4th edition, 2012

(Chapter 9, p. 335 – 399)

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Contents

1. Introduction

2. Finite distributed lags

3. Serial Correlation

4. Other tests for serially correlated errors

5. Estimation with serially correlated errors

6. Autoregressive distributed lag models

7. Forecasting

8. Multiplier analysis

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9.1 Introduction

• When modeling relationships between variables, the nature of the data

that have been collected has an important bearing on the appropriate

choice of an econometric model.

• For time series data:

o Time-series observations on a given economic unit, observed over a number

of time periods, are likely to be correlated.

o Time-series data have a natural ordering according to time.

• There is also the possible existence of dynamic relationships between

variables.

o A dynamic relationship is one in which the change in a variable now has an

impact on that same variable, or other variables, in one or more future time

periods.

o These effects do not occur instantaneously but are spread, or distributed,

over future time periods.

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9.1.1 Dynamic nature of relationships

How to model the dynamicnature of the relationship?

How can dynamic models be

used for forecasting?

• Distributed lag model: a dependent variable is a function of current and

past values of an explanatory variable: !" ! # "$"% $"!#% $"!$% & & & %.

• Autoregressive (distributed lag) model: a lagged dependent variable is

specified as one of the explanatory variables: !" ! # "!"!#% $"%, or, e.g.

" ! # "!"!#% $"% $"!#% $"!$%.• Autocorrelated errors: the impact of an unpredictable shock will be felt

in period " and in future periods: !" ! # "$"% & '"% '" ! ("'"!#%.

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9.1.2 Least squares assumptions (1)

• Dynamic models imply correlation between and or and or

both.

o When a variable is correlated with its past values, we say that it is

autocorrelated or serially correlated .

• As a consequence, the assumption MR4 is possibly violated:

• In a time series context, we use indexes and :

• Throughout this chapter, we maintain the assumption of stationarity

o A stationary variable is one that is not explosive, nor trending, and nor

wandering aimlessly without returning to its mean.

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9.2 Finite distributed lags (1)

• Consider a linear model in which, after time periods, changes in no

longer have an impact on .

o Note the notation change: is used to denote the coefficient of and

is introduced to denote the intercept.

• This model has two uses:

o Forecasting: given a sample for periods , what can we say

about period ?

Problem: How do we use the time series of -values to forecast ?

o Policy analysis: what is the impact on of a change in ?

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9.2 Finite distributed lags (2)

Interpretation of the effects

Assume that is increased by 1 unit and then returns to its original level.• Impact multiplier:

o The immediate effect will be an increase in by units.

• Distributed-lag weight or -period delay multiplier:

o will increase by units, …, will increase by units.

o In period , the value of returns to its original level.

• We have a finite distributed lag model:

o Distributed: the effect of a one-unit change in is distributed over thecurrent and next periods.

o Finite: after periods, changes in no longer have an impact on .

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9.2.1 Finite distributed lags: assumptions

• In distributed lag models, both and are typically random (we do not

set and then observe , they are observed together).

• Because the time series variables are random, we mention alternative

assumptions to obtain unbiased OLS estimators:

o ’s are random and is independent of all ’s (past, current, future).

o Normally distributed error terms! and tests can be used.

• Assumptions of the distributed lag model:

1. ( ).

2. and are stationary random variables, and is independent of all ’s.

3.

4. 5.

6.

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9.2.2 Finite distributed lags: Okun’s law (1)

• Okun’s Law: the change in unemployment rate ( ) from one period to

the next depends on the rate of growth ( ) of output in the economy:

(we expect )

where the constant is the “normal” growth rate (rate of output

growth needed to maintain a constant unemployment rate).

• We can rewrite this as:

where , and .

• Changes in output are likely to have a distributed lag effect on

unemployment:

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• We use quarterly U.S. data on unemployment and percentage change in

GDP from 1985Q2 to 2009Q3.

• Output growth is defined as:

• The series appear to be stationary.

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Interpretation of the coefficients (holding other factors fixed):

•

A 1% increase in growth rate leads to:o (impact) a fall in unemployment rate of 0.20 in the current quarter

o (delay) a fall of 0.16 in the next quarter

o (delay) a fall of 0.07 two quarters from now

• The effects of a sustained increase in growth rate of 1% are:

o (interim) -0.367 for one quarter

o (interim) -0.437 for two quarters (this is also the total multiplier)

• We estimate that the total effect of a change in output growth is

• An estimate of normal growth rate: , so the

estimated normal growth rate is 1.3% per quarter.

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9.3 Serial correlation

• When is assumption TSMR5, for likely to be

violated, and how do we assess its validity?

• When a variable exhibits

correlation over time, we say it

is autocorrelated or serially

correlated .

• Both observable time seriesvariables (like or ) and the

error can be autocorrelated.

• Questions:

o How do we measure

autocorrelation?

o How do we test for

autocorrelation?

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9.3.1 Serial correlation in output growth (1)

• Recall that the population correlation between two variables and is

given by:

• For the Okun’s Law problem, we have:

(Second equality holds because for stationary series)

• The notation is used to denote the population correlation between

observations that are one period apart in time.

o This is known also as the population autocorrelation of order one.

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• The first-order sample autocorrelation for G is obtained using the

estimates:

,

• Making the substitutions:

• More generally, the -th order sample autocorrelation for a series

that gives the correlation between observations that are periods apart:

• In our example, verify that , , , .

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• How do we test whether an autocorrelation is significantly different

from zero?

• The null hypothesis is , and )* can be estimated as +*.

• The following result holds in large samples when , ' is true:

- ! +* ! '

! #./ !"

/ +* # 0 "'% #%

• Example for the series 1 (with 2 ! '&'( and / ! )*):

- # !"

)* $ '&+)+ ! +&*) - $ !"

)* $ '&+## ! +&', - - !

" )* $ '(+ ! #&($ - + !

" )* $ '&$'' ! #&)*

o We reject the hypotheses , ! . )" ! ' and , ! . )# ! ' and conclude that 1

exhibits significant serial correlation at lags one and two.

• 95% confidence bounds on +* are #&)/."

/ and!#&)/."

/ .

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• The correlogram or sample autocorrelation function is the sequence

of autocorrelations +#% +$% +-% & & &

• It shows the correlation between observations that are one period apart,

two periods apart, three periods apart, and so on.

• Bounds: %#&)/."

/ ! %'&$'$.• Ljung-Box test (in gretl):

3&4 ! / "/ & $%4"

*$"

+#*"/ ! *% # 5

#4!6

is the maximal lag in the 3& statistic and 6 is the number ofparameters in the model.

In case there is no model,6 ! '.

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Partial autocorrelations

• Partial autocorrelations are used to measure the degree of associationbetween and , when the effects of other time lags

( ) – are removed

• The partial autocorrelation coefficient of order ( ) can be calculated

as the estimated coefficient in:

• Note: the first partial autocorrelation always equals the first

autocorrelation

• We use the same critical values to assess if the partial

autocorrelations are significantly different from zero

• Partial autocorrelations are plotted in a partial autocorrelation function

or PACF.

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9.3.2 Serially correlated errors (1)

• Consider a model for a Phillips Curve: 708 " ! 7 08 9

" ! : "; " ! ; "!#% o 708 9 " are inflationary expectations for period " (assumed constant)o is an unknown positive parameter

o Falling levels of unemployment (; " ! ; "!" < ') reflect excess demand forlabor! wages ! ! prices !

o Rising levels of unemployment (; "

!; "!" = ') reflect excess supply of

labor! wages " ! prices "

• We (initially) assume that inflationary expectations (708 9 " ) are constantover time (> # ! 708

9 " ), set > $ ! !: , and add an error term:

708 " ! > # & > $?; " & '"

• To determine if the errors are serially correlated, we compute the leastsquares residuals:

0'" ! 70 8 " ! @# ! @$?; "

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• We use quarterly Australian data from 1987Q1 to 2009Q3.

• We assume the series are stationary.

Price inflation Change in unemployment rate

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• The *-th order autocorrelationfor the residuals can be written

as (alternative formula):

+* !

/ #"!*

0'"0'"!*

/

#"!* 0'$"!

*

• The least squares equation is:

! 708 "12%

! '&,,,/"'&'/(*%

! '&($,)"'&$$)+%

?;

• The autocorrelation values at thefirst lags are:

+# ! '&(+), +$ ! '&+(/

• Correlogram:

!"

!#$%

#

#$%

"

# ' ( ) * "# "'

+,-

./0123,+ 456

7! "$8)9:;#$%

!"

!#$%

#

#$%

"

# ' ( ) * "# "'

+,-

./0123,+

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9.4.1 Other tests for serially correlated errors: LM (1)

• Assume a model !" ! > # & > $$" & '"

• We want to test whether errors that are one period apart are correlated,that is whether or not 345"'"% '"!#% is equal to zero.

• If we assume that '" ! )'"!# & A ", we can write the model as:" ! > # & > $$" & )'"!# & A " and test , ' . ) ! '.

• First approach: if '"!

# would be observable, we could use a "-test or 8 -test for this null hypothesis.

o Replace '"!" by the lagged LS residual 0'"!" and perform the test.o We need a value for 0'! ! (1) delete the first observation or (2) set 0'! ! '.

• Phillips example:

o Solution (1): " ! /&$#), 8 ! -*&/,, 657892 ! '&''' ! reject , ! . ) ! ' o Solution (2): " ! /&$'$, 8 ! -*&+,, 657892 ! '&''' ! reject , ! . ) ! '

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9.4.1 Other tests for serially correlated errors: LM (2)

• Second approach: use the goodness-of-fit statistic from an auxiliary

regression.

o Starting from the equation !" ! > " & > #$" & )0'"!" & A " and noting that!" ! @" & @#$" & 0'", we finally get: 0'" ! : " & : #$" & )0'"!" & A "

o If this regression has explanatory power, it will come from 0'"!".o If , ! . ) ! ' is true, then CD ! / $E# # 5#%"& o Again, we need a value for 0'! ! (3) discard or (4) set 0'! ! '.

• Phillips example (note that 5"'&)(:#% ! -&*+):

o Solution (3): CD ! "/ ! #% $ E# ! *) $ '&-#'$ ! $,&/# ! reject , ! o Solution (4): CD ! / $E# ! )' $ '&-'// ! $,&() ! reject , !

• For a four-period lag (note that 5"'&)(:+% ! )&+)):

o Solution (5): CD ! "/ ! +% $ E# ! */ $ '&-**$ ! --&+ ! reject , ! o Solution (6): CD ! / $E# ! )' $ '&+',( ! -/&, ! reject , !

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9.4.2 Other tests for serially correlated errors: DW (1)

• We want to test , ' . ) ! ' versus , # . ) = ' (positive autocorrelation)in the AR(1) error model , with .

• Estimate the regression by OLS and save 0'" ! !" ! 0!".

• The DW test statistic is: F !

/ #"$#

"0'"!0'"!"%#/ #

"$"0'#"

' $ ! $+#

• Tables provide critical values FC and F; o If F = F; , do not reject , ! (no first-order autocorrelation).

o If F < FC, conclude , " (positive first-order autocorrelation).

o If FC < F < F; , the test is inconclusive.

• Note: small values for F imply that 0'" is close to 0'"!# ! reject , !!

Values for F close to 2 imply no first-order autocorrelation.• DW is used less frequently today (because of availability in software

and problems when equation contains lagged dependent variable).

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9.5 Estimation with serially correlated errors

• If the assumption 345"'"% 'G% ! ' is violated, we want to know:

o What are the implications for OLS?

o How do overcome the negative consequences?

• Three estimation procedures are considered:

o Least squares estimation

o An estimation procedure that is relevant when the errors are assumed to

follow a first-order autoregressive model '" ! )'"!" & A " o A general estimation strategy for models with serially correlated errors

• We will encounter models with a lagged dependent variable, such as:

" ! H & I#!"!# & H '$" & H #$"!# & A ", which violate TSMR2.o We introduce an extra assumption “2” (TSMR2A) for models with a lagged

dependent variable:In the regression model !" ! > " & > #$"# & ( ( ( & > J $"J & A ", where some of the$"* may be lagged values of !, A " is uncorrelated with all $"* and their past values.

o LS estimator is then no longer unbiased, but remains consistent

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9.5.2 Estimating an AR(1) error model (1)

• Assume a model !" ! > # & > $$" & '", with '" ! )'"!# & A " [KE"#%].

o Properties of the random errors A " are:9 "A "% ! ', 57="A "% ! L#A and 345"A "% A G% ! ' for " )! G.

o We also assume that!# < )

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• Analogously to the sample correlogram +#% +$% & & & we can define apopulation correlogram as )#% )$% & & &

o For an AR(1) model, the population correlogram is )% )$% )-% & & &

o Since!# < )

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• Assume a model !" ! > # & > $$" & '", with AR(1) error representation'" ! )'"

!# & A ".

• We can write this model as

" ! > #"# ! )% & > $$" & )!"!# ! )> $$"!# & A " • Minimizing M A !

#/ "!$ A

$" (sum of squares of uncorrelated errors A ")

yields an estimator that is best and whose standard errors are correct in

large samples.• Properties

o Nonlinear model in the parameters (parameter for $"!" is!)> #)! requires nonlinear estimation (numerical methods).

o The model contains lagged ! and $ as RHS variables, so we drop the firstobservation (sum starts at ).

o Generalized least squares estimation can be used (Cochran-Orcutt procedure

and Prais-Winsten estimator).

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• Phillips curve (NLS estimation)

! 70 8 %'( ! '&,/')%!&"#)* ! '&/)++%!)+, ?; '" ! '&((,%!&!,! '"!" & A "

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9.5.2 Estimating an AR(1) error model: CO

• We can write the model as:

(1)

• First problem: is unknown (and needed to transform the variables)

• We know that

(2)

• The Cochrane-Orcutt estimation procedure is then:

o In (2), replace and with OLS estimates and and obtain an estimate

for .

o Use this estimate in (1) to transform the variables and obtain newestimates and for and .

o On convergence, we obtain the NLS estimator.

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9.5.2 Estimating an AR(1) error model: example

Cochrane-Orcutt estimation Prais-Winsten estimation

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9.5.3 Estimating a more general model

• We wrote the AR(1) error model as:

(1) " ! > #"# ! )% & > $$" & )!"!# ! )> $$"!# & A " • Consider the following autoregressive distributed lag model:

(2) " ! H & H '$" & H #$"!# & I#!"!# & A "

o We have the same variables, and one extra parameter (4 instead of 3)

o If A " have the right properties (zero mean, constant variance, uncorrelated),the model can be estimated by OLS (in large samples)

• Model (2) equals model (1) if , ' . H # ! !I#H ' is true.o We use a Wald test statistic to test this hypothesis (see gretl).

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9.5.3 Estimating a more general model: example (1)

• Estimating the more general model using the data for the Phillips curve

example yields:

! 708 ""12%

! '&---/"'&'*))%

& '&(()-"'&')'*%

708 "!# ! '&/**$"'&$+))%

?; " & '&-$''"'&$(,(%

?; "!#

• The equivalent AR(1) estimates are (see NLS solution):

o 0H ! 0> ""#

! 0)% ! '&,/')

$"#

!'&((,+% ! '&--/* (compare with 0.3336)

o 0I" ! 0) ! '&((,+ (compare with 0.5593)

o 0H ! ! 0> # ! !'&/)++ (compare with -0.6882)o 0H " ! !0) 0> # ! !'&((,+ $ "!'&/)++% ! '&-*,# (compare with 0.3200)

• Test the restriction , ' . H # ! !I#H ': 5$"#% ! '#$, with B-value 0.738.We cannot reject , ' and conclude that the AR(1) model is not toorestrictive.

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9.5.3 Estimating a more general model: example (2)

• The original economic model for the Phillips Curve was:

708 " ! 708

9

" ! : "; " !; "!#% • In the more general model, we now have:

o An estimate of the model for inflationary expectations:

70 8 9 " ! '&---/ & '&(()-70 8 "!"

o An estimate of the effect of unemployment:

!'&/**$"; " !; "!"% & '&-$''"; "!"!; "!#% • The coefficient of ?; "!# is not significantly different from zero.

! 708 ""12%

! '&-(+*"'&'*,/%

& '&($*$"'&'*(#%

708 "!# ! '&+)')"')$#%

?; "

o Inflationary expectations are now 708 9 " ! '&-(+* & '&($*$708 "!#,o A 1% rise in unemployment rate leads to an approximate 0.5 fall in inflation

rate.

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9.6 Autoregressive Distributed Lag Models (1)

• An autoregressive distributed lag (ARDL) model is one that contains

both lagged $"’s and lagged !"’s.

• ARDL( , N ) = ARDL with B lags in ! and N lags in $:

• Two examples:

o ARDL(1, 1): ! 708 " ! '&---/ & '&(()-708 "!" ! '&/**$?; " & '&-$''?; "!" o ARDL(1, 0): ! 708 " ! '&-(+* & '&($*$708 "!"! '&+)')?; "

• An ARDL model can be transformed into one with only lagged $’swhich go back into the infinite past (infinite DL model):

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9.6 Autoregressive Distributed Lag Models (2)

• OLS is an appropriate estimation technique under the assumption

TSMR1, TSMR2A, TSMR3-5.

• How do we choose B and N ? There are 4 possible criteria:

o Has serial correlation in the errors been eliminated? If not, OLS will be

biased in small and large samples.

o Are the signs and magnitudes of the estimates consistent with our

expectations from economic theory?

o Are the estimates significantly different from zero, particularly those at the

longest lags?

o What values for B and N minimize information criteria like AIC and SC?

K7O ! 8<

$MM9

/ %&

$J

/ MO ! 8<

$MM9

/ %&

J 8

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9.6.1 The Phillips curve (1)

Example: Phillips curve

•

Consider the previously estimated ARDL(1,0) model (see slide 38): ! 708 "

"12%! '&-(+*

"'&'*,/%& '&($*$

"'&'*(#%708 "!# ! '&+)')

"')$#%?; ", obs = 90

-values for LM test for

autocorrelation

Correlogram for the residuals from

ARDL(1,0) model

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9.6.1 The Phillips curve (2)

• For an ARDL(4,0) version of the model (obs = 87):

! 708 "%'(& ! '''#%!&!,-.& &'&$-(+%!&"!"/&708 "!" & '$#-%!&"!.-&708 "!# & '/,,%!&"!*!&708 "!.

&'&$*#)%!&"!")&

708 "!) ! '&,)'$%!&"--*&

?; "

• Inflationary expectations are given by:

708 9 " ! '''# &'&$-(+708 "!" & '$#-708 "

!#

&'/,,708 "!. & '&$*#)708 "!)

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9.6.2 Okun’s Law

Example: Okun’ Law

•

Recall the model for Okun’s Law (obs = 96), see slide 13:&?; "%'(&

! '&(*-/%!&!)+#&

! '&$'$'%!&!.#)&

1" ! '/(-%!&!..*&

1"!" ! '&',''%!&!.."&

1"!#

• Now consider this ARDL(1,1) version:&?; ""12%

! '&-,*'"'&'(,*%

& '&-('#"'&'*+/%

?; "!# ! '*+#"'&'-',%

1" ! '&'))$"'&'-/*%

1"!#

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9.6.3 Autoregressive models

• An autoregressive model of order B, denoted AR( B), is given by:

• Consider an AR(2) model for growth in real GDP:

'1"%'(&

! '&+/(,%!&")..&

& '&-,,'%!&"!!!&

1"!" & '&$+/$%!&"!#,&

1"!# (obs = 96)

• Question: consider the correlogram of 1". What do you observe?

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9.7.1 Forecasting with an AR model (2)

• The forecast error at time / & #, / & $ and / & -:

Q# ! 1/ ! 01/ ! "H !

0H % & "I# !

0I#%1/ & "I$ !

0I$%1/ !# & A / ! A /

Q$ ! I#"1/ ! 01/ % & A / &$ ! I#Q# & A / &$! I#A / & A / &$

Q- ! I#Q$ & I$Q# & A / &-! "I$# & I$%A / & I#A / &$ & A / &-

• Because the A "’s are uncorrelated with constant variance L$A , we have:

L$# ! 57="Q#% ! L$A

L$$ ! 57="Q$% ! L$A "# & I

$#%

L$- ! 57="Q-% ! L$A ""I

$# & I$%

$ & I$# & #%

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9.7.2 Forecasting with an ARDL model

• Consider forecasting ; " using the Okun’s Law ARDL(1,1):

?; " ! H & I#?; "!# & H '1" & H #1"!# & A " • The value of ?; in the first post-sample quarter is:

?; / ! H & I#?; / & H '1/ & H #1/ & A /

o We need a value for 1/ 0" (e.g. independent forecasts, what-if questions)

• Now consider the change in unemployment

; / ! ; / ! H & I#"; / ! ; / !#% & H '1/ & H #1/ & A / Rearranging:

; / ! H & "I# & #%; / ! I#; / !# & H '1/ & H #1/ & A / ! H & I

;

/ & I

&$;

/ !# & H

'1

/ & H

#1

/ & A

/

which is an ARDL(2,1) model for the level of unemployment.

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9.7.3 Exponential smoothing (1)

• Another popular model used for predicting the future value of a variable

on the basis of its history is the exponential smoothing (ES) method.

• Like forecasting with an AR model, forecasting using ES does notutilize information from any other variable.

• One possible forecasting method is to use average past information, e.g.:

o This is an example of a simple (equally-weighted) moving average modelwith .

• Consider a form in which the weights decline exponentially as the

observations get older:

o We assume that .

o Also, it can be shown that the weights sum to one: .

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• For forecasting, an infinite sum is not convenient. Recognizing that:

we can simplify to: .

o The next period forecast is a weighted average of the forecast for the current

period and the actual realized value in the current period.

• The value of can reflect one’s judgment about the relative weight of

current information.

o It can be estimated from historical information by obtaining within-sample

forecasts:

, for

o Next, choose that minimizes the sum of squares of the one-step forecasterrors:

.

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• Example: exponential smoothing forecasts for GDP growth

• The forecasts for 2009Q4 from each value of are:

o :

o :

9 8 M l i li l i (1)

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9.8 Multiplier analysis (1)

• Multiplier analysis refers to the effect, and the timing of the effect, of a

change in one variable on the outcome of another variable

• Let’s find multipliers for an ARDL( ,N ) model of the form:

• We can transform this into an infinite distributed lag model:

• The multipliers are defined as:

o s-period delay multiplier:

o s-period interim multiplier:

o Total multiplier:

9 8 M l i li l i (2)

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• The lag operator is defined as: or, in general

• Now rewrite our model as:

• Rearranging terms:

• Okun’s Law example: the model

?; " ! H & I#?; "!# & H '1" & H #1"!# & A "

can be rewritten as:

"# ! I#C%?; " ! H & "H ' & H #C%1" & A "

9 8 M lti li l i (3)

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• Define the inverse of as such that:

• Then we have:

?; " ! "#! I#C%!#H & "# ! I#C%!#"H ' & H #C%1" & "# ! I#C%!#A " • Equating this with the infinite distributed lag representation:

• For these equations to be identical, it must be true that:


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• Multiply both sides of equation (1) by : .

• The lag of a constant is the same constant: .

• Now we have and .

• Multiply both sides of equation (2) by :

• Rewrite equation (2) as:

• Equating coefficients of like powers in yields: , ,

, , and so on.


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• We can now find the ’s using the recursive equations:

• You can – more easily – start from the equivalent of equation (4) which,

in its general form, is:

o Given the values and for your ARDL model, you need to multiply out

the above expression, and then equate coefficients of like powers in the lag

operator.

• Reconsider the Okun’s Law model:

?; " ! '&-,*' & '&-('#?; "!# ! '*+#1" ! '&'))$1"!#


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• The impact and delay multipliers for the first four quarters are:


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• We can estimate the total multiplier given by and the normal

growth rate that is needed to maintain a constant rate of unemployment

by .

• We can show that:

• An estimate for is given by:

• Therefore, normal growth rate is:

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ABE Ch09 Hill