Dynamic Models

Introduction

• Assess the partial adjustment model as an example of a dynamic model

• Examine the ARDL model as a means of testing the partial adjustment model

• Discuss the Sharpe-Lintner model

• Derive the Error Correction Model from the ARDL model.

Partial Adjustment Model

• This type of dynamic model is part of the Koyck distribution class of models.

• It is used in models where adjustment does not occur immediately, but takes a number of time periods to fully adjust.

• We can apply a specific restriction to a general ARDL model to determine if partial adjustment is taking place

Partial Adjustment Model

• This model has as its dependent variable a desired value or planned value.

• This desired value is then determined by the usual explanatory variables:

tt

ttt

yy

uxy

of level desired *

10*

Partial Adjustment Mechanism

• The partial adjustment mechanism takes the following form, where the change in y is equal to the difference between the desired value of y and its actual value in the previous time period:

10

tcoefficien adjustment of speed the

)( 1*

1

tttt yyyy

Partial Adjustment Mechanism

• If λ is 0, it means there is no adjustment from one time period to the next.

• If λ is 1, it means there is immediate adjustment from one time period to the next.

• To produce an estimating equation, we need to substitute in this adjustment mechanism, to have an expression in terms of y as the dependent variable rather than y*.

Estimating Equation

• The estimating equation, where y is the dependent variable, following the process of substitution is:

tttt uxyy 110 )1(

ARDL Model

• The estimating equation is the same as the conventional ARDL model, including a specific restriction. The restriction is that the coefficient on the x(t-1) variable is equal to 0. Where the ARDL model is:

ttttt vxxyy 132110

Partial Adjustment Model

• In addition the estimated coefficients on the other variables can be used to produce estimates of the coefficients in the original model with the ‘desired’ variable. Based on the previous ARDL model:

00

12

11 )1()1(

Partial Adjustment Model

• An example of the partial adjustment model is Lintner’s Dividend-Adjustment Model.

• The model suggests that dividends are related to company profits, but when profits rise, dividends do not rise in the same proportion immediately.

• Lintner suggests that firms have a long-run desired/ target pay-out ratio between dividends and profits, in which the dividend payout relative to profits is a desired level.

ratiopayout

*

ttt uD

Lintner Model

• Lintner then estimated the following model for the US:

profits

dividendsD

DD ttt

170.015.03.352ˆ

Lintner Model

• The coefficient on D(t-1) is equal to (1-λ)=0.70.

• This means that the speed of adjustment λ=(1-0.70)=0.30. This suggests adjustment is relatively slow.

• The coefficient on π is equal to (β λ), so β =0.15/0.30= 0.50. This gives a value for the payout ratio of 0.5.

Error Correction Models

• The error correction model is a short-run dynamic model, consisting of differenced variables, except the error correction term.

• The error correction term reflects the difference between the dependent and explanatory variable, lagged one time period.

• This model can incorporate a number of lags on both the dependent and explanatory variables.

Error Correction Models

• As with the partial adjustment model, the error correction model (ECM) can be interpreted as a general ARDL model, in which a specific restriction is applied.

• The model can be derived from the basic ARDL model:

ttttt vxxyy 132110

Error Correction Model

• To turn the ARDL model into the ECM involves the following:

- Subtract y(t-1) from both sides of the ARDL equation.

- Add β(2)x(t-1) and subtract the same amount from the right hand side of equation.

- Collect terms.

Error Correction Model

• The previous rearrangements produces the following model:

ttttt vxyxy 1321120 )()1(

Error Correction Model

• Finally to produce an ECM, the following restriction needs to be applied to the lagged level variables:

1

)(1

321

321

Error Correction Term

• The ECM is usually written in the following form, where the parameter τ is the error correction term coefficient.

ttttt uxyxy )( 1120

Long-Run

• In the long-run, we assume the Δy and the Δx variables (all variables in logs) grow at a constant rate of g. This produces the following expression:

*)1(

*

*)*(

20

20

xg

y

xygg

Long-run

• If we assume the relationship between x and y is of the form: y*=k.x*, taking logs gives: log(y*)=log(k)+log(x*). To find k we need to antilog the previous expression:

])1(

exp[ 20

g

k

ECM

• The following estimates were produced for an ECM using 60 observations (All variables in logs), where the long-run growth rate of Δx and Δy is 0.02:

(0.2) (0.02) (0.1)

)(5.02.07.0ˆ 11 tttt xyxy

ECM

• This produces the following long-run relationship between x and y, which takes the form y = k. x :

93.3

]5.0

684.0exp[]

5.0

02.0)12.0(7.0exp[

k

ECM

• The ECM is used to model the short-run in many situations and is closely associated with cointegration (we will cover this later).

• The model usually also includes lagged variables in addition to the error correction term.

Conclusion

• The Partial Adjustment model is a version of the Koyck distribution

• It provides a theoretical reason for the inclusion of lags in a model.

• The Error Correction Model (ECM) can be derived from an ARDL model.

• The estimates from an ECM can also be used to determine the long-run relationship between variables.