Distributed lags

Distributed lags are dynamic relationships in which the effectsof changes in some variable X on some other variable Y are spread through time.

Distributed lags can arise for a variety of reasons including:

1. Costs of adjustment2. The effects of expectations

Example: A dynamic consumption function

This is a difference equation of the form:

11.79 0.1947 0.8016t t tlc ly lc

Dynamic effects of an increase in disposable income

Time LY LC0 0 0.00001 1 0.19472 1 0.35083 1 0.47594 1 0.57625 1 0.65666 1 0.72107 1 0.77268 1 0.81419 1 0.847210 1 0.8739

The Costs of Adjustment Model

2 2*1

*

1

2 2t t t t

t t t

C Y Y Y Y

Y X u

Quadratic adjustment costs penalise large deviations morestrongly than small deviations.

C z( ) 0.5 z2

10 5 0 5 10

20

40

60

C z( )

z

Quadratic costs of adjustment give rise to the partial adjustmentmodel.

*1

*1

1

0

1

1

1 1

tt t t t

t

t t t

t t t t

dCY Y Y Y

dY

Y Y Y

Y X Y u

This provides a rationale for the introduction of lagged endogenous variables into regression models.

Backward substitution yields:

2

1 22 3

2

1 32 3

....1 1 1

1...

1 1 1

t t t t

t t t

Y X X X

u u u

The effects of a change in X on Y are spread out over time. Theweights on past values of X decline for longer lags because:

1 if 0 <11

The long run effect of a change in X on Y can be calculatedusing the following formula:

1

1

1

1

1

/ (1 )lim lim

1 / (1 )1

it

ii

it

it ti

g t

g t

For this limit to converge we need

1

We also usually assume θ >0 for a sensible economic model.

Example:

Parameters

1 0.5

Weights Cumulative Effect of X on Y

g t( )

1

t

i

i 1

1 i

f i( )

i 1

1 i

i 1 10 t 1 10

0 5 10

0.5

1

f i( )

i

0 5 100.6

0.8

1

g t( )

t

Example: An accelerator model of investment

In this case investment is determined by the following differenceequation:

1 2 1ln 0.49 1.74ln 1.07 ln 0.54ln 0.89lnt t t t tI Y Y Y I

The long run effect of an increase in GDP on investment canbe determined from the following expression:

1.74 1.07 0.541.18

1 0.89