Lags and Lag Structure

8/2/2019 Lags and Lag Structure

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Lags and Lag Structure

Distributed Lag Modeling



Introduction

• We are often interested in variables that change

across time rather than across individuals.

• Simple Static models relate a time series variable

to other time series variables.

• The effect is assumed to „operate‟ within the

period.



Dynamic Models

• Dynamic effects

– Policy takes time to have an effect.

– Size and nature of effect can vary over time.

– Permanent vs. Temporary effects.



The „Principle‟

• Macroeconomics

– e.g. the effect of X on Y in short run vs. the long run

• impulse response function

– X increases by 1 in year 1

– returns to normal afterwards

– what happens to y over time?

time

Y



Distributed Lag

• Effect is distributed through time

– consumption function: effect of income through time

– effect of income taxes on GDP happens with a lag

– effect of monetary policy on output through time

yt = + 0 xt + 1 xt-1 + 2 xt-2 + et

it

t

i x

y E

)(



The Distributed Lag Effect

Economic action

at time t

Effect

at time t

Effect

at time t+1

Effect

at time t+2



The Distributed Lag EffectEffect at time t

Economic action

at time t

Economic action

at time t-1

Economic action

at time t-2



Two Questions

1. How far back?

- What is the length of the lag?

- finite or infinite

2. Should the coefficients be restricted?

- e.g. smooth adjustment

- let the data decide



Unrestricted Finite DL

• Finite: change in variable has an effect on another only for

a fixed period

– Fare Price Changes affect Ridership for 6 months

– the interval is assumed known with certainty

• Unrestricted (unstructured)

– the effect in period t+1 is not related to the effect in period t



yt = + 0 xt + 1 xt-1 + 2 xt-2 + . . . +n xt-n + et

n unstructured lags

no systematic structure imposed on the ‟s

the ‟s are unrestricted

OLS will work

produce consistent and unbiased estimates



Problems

1. n observations are lost with n-lag setup.

• data from 1990, 5 lags in model implies earliest point in regression is 1995

• use up degrees of freedom (n-k)

2. high degree of multicollinearity among xt-j‟s

• xt is very similar to xt-1 --- little independent information

• imprecise estimates

• large std errors, low t-tests

• hypothesis tests uncertain.

3. Several LHS variables• many degrees of freedom used for large n.

4. Could get greater precision using structure



Arithmetic Lag Structure

i

i

0 = (n+1)

1 = n

2 = (n-1)

n =

.

.

.

0 1 2 . . . . . n n+1

..

.

.

linear

lag

structure

1. effect of X eventually goes to zero

2. Effect of each lag will be less than previous of



The Arithmetic Lag Structure

Imposing the relationship:

ii = (n - i+ 1)

0 = (n+1)

1 = n

2

= (n-1)

3 = (n-2)

n-2 = 3

n-1 = 2 n =

only need to estimate one coefficient, ,

instead of n+1 coefficients, 0 , ... , n .

yt = + 0 xt + 1 xt-1 + 2 xt-2 + . . . +n xt-n + et



• Suppose that X is (log of) fare price and Y is (log of)

Ridership, n=12 and is estimated to be 0.1

• the effect of a change in x on Ridership in the current

period is 0(n+1)=1.3

• the impact of fare price changes one period later has

declined to 1n=1.2

• n periods later, the impact is n 0.1

• n+1 periods later the impact is zero

it

t i

x

y E

)(



Estimation

• Estimate using OLS

• only need to estimate one parameter:

• Have to do some algebra to rewrite the model inform that can be estimated.



Advantages/disadvantages

• Fewer parameters to be estimated (only one) thanin the unrestricted lag structure

– Lower standard errors

– Higher t-statistics

– More reliable hypothesis tests

• What if the restriction is untrue?

– Biased and inconsistent

– A bit like wrong exclusion restrictions in 2SLS• Is the linear restriction likely to be true?

– Look at unrestricted model

– Do f-test



2

1

/

/ )(

df SSE

df SSE SSE

F U

U R

F-test

• estimate the unrestricted model

• estimate the restricted (arithmetic lag) model

• calculate the test statistic

• compare with critical value F(df1,df2)

– df1=n the number of restrictions

• number of betas less number of gammas = (n+1)-1

– df2=number of observations-number of variables in the

unrestricted model (incl. constant)

– df2=(T-n)-(n+2)



Polynomial Distributed Lag

• Linear shape of lags maybe restrictive

• Want “hump” shape

• Polynomial --- quadratic or higher

i = 0 + 1i + 2i2

i

it

t

x

y E

)(



Polynomial Lag Structure

.. . .

.

0 1 2 3 4 i

i

0

12 3

4

• Similar idea to Arithmetic DL model; different shape to lags• Still Finite: the effect of X eventually goes to zero

• The coefficients are related to each other

• the effect of each lag will not necessarily be less than previous one i.e.

not uniform decline



Estimation

• Estimate using OLS

• only need to estimate p parameters:

– number of parameters is equal to degree of polynomial

• Have to do some algebra to rewrite the model inform that can be estimated.

– model reduces to arithmetic model if polynomial is of

degree 1

• Do OLS on transformed model



Advantages/Disadvantages

• Fewer parameters to be estimated (only the degree of

polynomial) than in the unrestricted lag structure

– more precise

• What if the restriction is untrue?

– biased and inconsistent

• Is the polynomial restriction likely to be true?

– more flexible than arithmetic DL

• what if approximately true?



Lag Length

• For all three finite models we need to choose the lag length(DL,ADL,PDL)

• Think of this as choosing the cut-off point

– The time beyond which a variable will cease to have an impact

• No satisfactory objective criterion for deciding this• Choose n to be infinite?



Lag-Length Criteria

• Akaike‟s AIC criterion

• Schwarz‟s SC criterion

• For each of these measures we seek that lag length that minimizes thecriterion used. Since adding more lagged variables reduces SSE , thesecond part of each of the criteria is a penalty function for addingadditional lags.

2( 2)ln n

SSE n AIC

T N T N

2 ln( )( ) ln n

n T N SSE SC n

T N T N



Key Points1. How far back?

- What is the length of the lag?

- No conclusive answer

2. Should the coefficients be restricted?

- Let the data decide: unrestricted

- Arithmetic or polynomial- What degree of polynomial



Geometric Lag Model

• DL is infinite --- infinite lag length

• But cannot estimate an infinite number of parameters

• restrict the lag coefficients to follow a pattern – estimate the parameters of this pattern

• For the geometric lag the pattern is one of continuous

decline at decreasing rate



Geometric Lag Structure

i

.

..

. .0 1 2 3 4 i

1 =

2 = 2

3 =

3

4 = 4

0 = geometrically

declining

weights



Estimation

• Cannot Estimate using OLS

– OLS will be inconsistent (just as with simultaneous equations)

• Only need to estimate two parameters: ,

• Have to do some algebra to rewrite the model in form thatcan be estimated.

• Then apply Koyck transformation

• Then use 2SLS

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Lags and Lag Structure