PARTIAL ADJUSTMENT 1 The idea behind the partial adjustment model is that, while a dependent variable Y may be related to an explanatory variable X, there

PARTIAL ADJUSTMENT

1

The idea behind the partial adjustment model is that, while a dependent variable Y may be related to an explanatory variable X, there is inertia in the system.

ttt uXY 21*

2

The actual value of Yt is a compromise between its value in the previous time period, Yt–1, and the value justified by the current value of the explanatory variable.

PARTIAL ADJUSTMENT

ttt uXY 21*

3

Let us denote the justified value of Y (or target, desired, or appropriate value, however you want to describe it) as Yt*, given by the equation shown.

PARTIAL ADJUSTMENT

ttt uXY 21*

4

In the partial adjustment model it is assumed that the actual increase in the dependent variable from time t – 1 to time t, Yt – Yt–1, is proportional to the discrepancy between the justified value and the previous value, Yt* – Yt–1.

PARTIAL ADJUSTMENT

1*1 tttt YYYY ttt uXY 21

*

5

l is usually described as the speed of adjustment.

PARTIAL ADJUSTMENT

1*1 tttt YYYY ttt uXY 21

*

6

The actual value in the current time period is therefore a weighted average of the desired value and the previous actual value. l logically should lie in the interval 0 (no change at all) to 1 (full adjustment in the current time period).

PARTIAL ADJUSTMENT

1*1 tttt YYYY

1* 1 ttt YYY

ttt uXY 21*

7

Substituting for Yt* from the original relationship, one obtains a regression specification in terms of observable variables of the ADL(1,0) form.

.1,, 32211

PARTIAL ADJUSTMENT

1*1 tttt YYYY

1* 1 ttt YYY

ttt uXY 21*

ttt

ttt

tttt

uYX

uYX

YuXY

1321

121

121

1

1

where

8

It follows that its dynamics are those of the ADK(1,0) model discussed in the previous slideshow. The short-run impact of X on Y is given by the coefficient b2 = g2l.

PARTIAL ADJUSTMENT

1*1 tttt YYYY

1* 1 ttt YYY

ttt uXY 21*

ttt

ttt

tttt

uYX

uYX

YuXY

1321

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121

1

1

9

The long-run effect can be evaluated by finding the relationship between the equilibrium values of Y and X.

PARTIAL ADJUSTMENT

1*1 tttt YYYY

1* 1 ttt YYY

YXY 121

ttt uXY 21*

ttt

ttt

tttt

uYX

uYX

YuXY

1321

121

121

1

1

10

1*1 tttt YYYY

1* 1 ttt YYY

YXY 121

XY 21

XY 21

The long-run effect turns out to be g2. This makes sense, since this is the coefficient in the equation determining the desired value of Y.

ttt uXY 21*

ttt

ttt

tttt

uYX

uYX

YuXY

1321

121

121

1

1

PARTIAL ADJUSTMENT

11

Brown's Habit Persistence Model of the aggregate consumption function was an early example of the use of a partial adjustment model. Desired consumption is related to wage income, nonwage income and a dummy variable.

PARTIAL ADJUSTMENT

tttt uANWWC 321*

12

The reason for separating income into wage income and nonwage income is that the marginal propensity to consume is likely to be higher for wage income than for nonwage income.

PARTIAL ADJUSTMENT

tttt uANWWC 321*

13

Brown fitted the model with a time series which included observations before and after the Second World War. The dummy variable, A, was defined to be 0 for the prewar observations and 1 for the postwar ones.

PARTIAL ADJUSTMENT

tttt uANWWC 321*

14

As the name of his model suggests, Brown hypothesized that there was a lag in the response of consumption to changes in income and he used a partial adjustment model.

PARTIAL ADJUSTMENT

tttt uANWWC 321* 1*

1 tttt CCCC 1

* 1 ttt CCC

15

Substituting for desired consumption, one obtains current consumption in terms of current income and previous consumption.

PARTIAL ADJUSTMENT

tttt

ttttt

uACNWW

CuANWWC

1321

1321

1

1

tttt uANWWC 321* 1*

1 tttt CCCC 1

* 1 ttt CCC

16

Brown fitted the model with aggregate Canadian data for the years 1926–1949, omitting the years 1942–1945, using a simultaneous equations estimation technique. The variables were measured in billions of Canadian dollars at constant prices. t statistics are in parentheses.

PARTIAL ADJUSTMENT

tttt uANWWC 321* 1*

1 tttt CCCC 1

* 1 ttt CCC

tttt

ttttt

uACNWW

CuANWWC

1321

1321

1

1

ACNWWC tttt 69.022.028.061.090.0ˆ1

(7.4) (4.2) (2.8)(4.8) (4.8)

17

The short-run marginal propensities to consume out of wage and nonwage income are 0.61 and 0.28, respectively. Note that the former is indeed larger than the latter. How would you test whether the difference is significant?

PARTIAL ADJUSTMENT

tttt uANWWC 321* 1*

1 tttt CCCC 1

* 1 ttt CCC

tttt

ttttt

uACNWW

CuANWWC

1321

1321

1

1

ACNWWC tttt 69.022.028.061.090.0ˆ1

(7.4) (4.2) (2.8)(4.8) (4.8)

18

The coefficient of lagged consumption literally implies that, if consumption in the previous year had been 1 billion dollars greater, consumption this year would have been 0.22 billion dollars greater.

PARTIAL ADJUSTMENT

tttt uANWWC 321* 1*

1 tttt CCCC 1

* 1 ttt CCC

tttt

ttttt

uACNWW

CuANWWC

1321

1321

1

1

ACNWWC tttt 69.022.028.061.090.0ˆ1

(7.4) (4.2) (2.8)(4.8) (4.8)

19

That is a bit clumsy. It is better to interpret it with reference to l in the adjustment process. It implies that the speed of adjustment is 0.78, meaning that 0.78 of the difference between desired and actual consumption is eliminated in one year.

PARTIAL ADJUSTMENT

tttt uANWWC 321* 1*

1 tttt CCCC 1

* 1 ttt CCC

tttt

ttttt

uACNWW

CuANWWC

1321

1321

1

1

ACNWWC tttt 69.022.028.061.090.0ˆ1

(7.4) (4.2) (2.8)(4.8) (4.8)

20

With the speed of adjustment, we can derive the long-run propensities to consume. We do this by dividing the short-run propensities by l. We find that the long-run propensity to consume out of wages is 0.78.

PARTIAL ADJUSTMENT

tttt uANWWC 321* 1*

1 tttt CCCC 1

* 1 ttt CCC

tttt

ttttt

uACNWW

CuANWWC

1321

1321

1

1

ACNWWC tttt 69.022.028.061.090.0ˆ1

(7.4) (4.2) (2.8)(4.8) (4.8)

78.022.0161.0

2

g 36.022.0128.0

3

g

21

ACNWWC tttt 69.022.028.061.090.0ˆ1

Similarly, the long-run propensity to consume nonwage income is 0.36. Note that, in this example, there is not a great difference between the short-run and long-run propensities. That is because the speed of adjustment is rapid.

(7.4) (4.2) (2.8)(4.8)

tttt uANWWC 321* 1*

1 tttt CCCC 1

* 1 ttt CCC

tttt

ttttt

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1321

1321

1

1

(4.8)

78.022.0161.0

2

g 36.022.0128.0

3

g

PARTIAL ADJUSTMENT

============================================================Dependent Variable: LGHOUS Method: Least Squares Sample(adjusted): 1960 2003 Included observations: 44 after adjusting endpoints ============================================================ Variable CoefficientStd. Errort-Statistic Prob. ============================================================ C 0.073957 0.062915 1.175499 0.2467 LGDPI 0.282935 0.046912 6.031246 0.0000 LGPRHOUS -0.116949 0.027383 -4.270880 0.0001 LGHOUS(-1) 0.707242 0.044405 15.92699 0.0000============================================================R-squared 0.999795 Mean dependent var 6.379059Adjusted R-squared 0.999780 S.D. dependent var 0.421861S.E. of regression 0.006257 Akaike info criter-7.223711Sum squared resid 0.001566 Schwarz criterion -7.061512Log likelihood 162.9216 F-statistic 65141.75Durbin-Watson stat 1.810958 Prob(F-statistic) 0.000000============================================================

22

Here is the result of a parallel logarithmic regression of expenditure on housing on DPI and relative price, using the Demand Functions data set.

PARTIAL ADJUSTMENT

23

The short-run income elasticity is 0.28.

PARTIAL ADJUSTMENT


24

The short-run price elasticity is 0.12. Both of these elasticities are very low. This is because housing is a good example of a category of expenditure with slow adjustment.

PARTIAL ADJUSTMENT


25

The adjustment rate implicit in the coefficient of LGHOUS(–1) is only 0.29. People do not change their housing quickly in response to changes in income and price. If anything, the estimated rate seems a little high.

PARTIAL ADJUSTMENT


26

The long-run income elasticity is 0.97, not far off the income elasticity in the static model in the first sequence for this chapter, 1.03.

PARTIAL ADJUSTMENT


97.07072.012829.0

long-run income elasticity


27

The long run price elasticity is 0.40, again not far from the estimate in the static model, 0.48. In this example the long-run elasticities are much greater than the short-run ones because the speed of adjustment is slow.

long-run price elasticity 40.07072.011169.0

PARTIAL ADJUSTMENT

Copyright Christopher Dougherty 2013.

These slideshows may be downloaded by anyone, anywhere for personal use.

Subject to respect for copyright and, where appropriate, attribution, they may be

used as a resource for teaching an econometrics course. There is no need to

refer to the author.

The content of this slideshow comes from Section 11.4 of C. Dougherty,

Introduction to Econometrics, fourth edition 2011, Oxford University Press.

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2013.01.20

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PARTIAL ADJUSTMENT 1 The idea behind the partial adjustment model is that, while a dependent variable Y may be related to an explanatory variable X, there