Adaptive expectations Adaptive expectations and and

partial adjustment partial adjustment

Presented by:Presented by:Monika TarsalewskaMonika TarsalewskaPiotrek Jeżak Piotrek Jeżak Justyna KoperJustyna KoperMagdalena PrędotaMagdalena Prędota

Adaptive expectationsAdaptive expectations

ExpectationsExpectations

Either the dependent variable or one of the Either the dependent variable or one of the independent variables is based on expectations. independent variables is based on expectations. ExpectationsExpectations about economic events are usually about economic events are usually formed by aggregating new information and past formed by aggregating new information and past experience. Thus, we might write the expectation of experience. Thus, we might write the expectation of a future value of variable a future value of variable xx, formed this period, as, formed this period, as

Example: Example: Forecast of prices and income enter Forecast of prices and income enter demand equation and consumption equations. demand equation and consumption equations.

.,,,,\ ,...21,...21*

1 ttttttttt xxzgxxzxEx

Adaptive expectationsAdaptive expectations

Regression:Regression:

The error of past observation:The error of past observation:

and a mechanism for the formation of the expectationand a mechanism for the formation of the expectation::

)1(*1 ttttt wxy

211 *1

*1

*1 tttttttt xLxxxx

*1

*1

*1 1 ttttttt xxxx

Adaptive expectationsAdaptive expectations

The expectation variable can be written asThe expectation variable can be written as

Inserting equation (3) into (Inserting equation (3) into (11) produces the ) produces the geometric distributed lag model.geometric distributed lag model.

3...11

12

21

*1

tttttt xxxxL

x

4...1 22

1 tttttt wxxxy

Adaptive expectationsAdaptive expectations

Koyck transformationKoyck transformation

ttttttt wxxxxy ...1 33

22

1

1133

22

11 ...1 tttttt wxxxy

111 )(11 ttttttt wwxyy

There is a problem of There is a problem of simultaneity simultaneity as as yyt-1t-1 is correlated is correlated

in time within time with

There is nonlinear restriction in our model which There is nonlinear restriction in our model which should de included in the regressionshould de included in the regression

Adaptive expectationsAdaptive expectations

1*

tttu

***

1

*

1 111

tu

wwxyy ttttttt

*

0*

tp

tt inccons

ptt

pt

pt incincincinc 11 1

And after Koyck transformation

10

11 11 ttttt incconscons

Measurement of permanent income might be approached through the use of the adaptive expectations hypothesis, where permanent income (inct) alters between periods in proportion to the difference between actual income (inct) in a period, and permanent income in previous period.

Adaptive expectationsAdaptive expectations

ivreg conspr (l.conspr = l2.conspr l3.conspr l4.conspr) housedisp

Instrumental variables (2SLS) regression

Source | SS df MS Number of obs = 10-------------+------------------------------ F( 2, 7) = 4419.15 Model | 14.1834892 2 7.09174462 Prob > F = 0.0000 Residual | .011197658 7 .001599665 R-squared = 0.9992-------------+------------------------------ Adj R-squared = 0.9990 Total | 14.1946869 9 1.57718743 Root MSE = .04

------------------------------------------------------------------------------conspr | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+----------------------------------------------------------------conspr | L1 | .5213197 .0819684 6.36 0.000 .3274953 .7151441housedisp | .4497056 .0927137 4.85 0.002 .2304726 .6689387_cons | .2452798 .1028115 2.39 0.048 .0021692 .4883904------------------------------------------------------------------------------

Adaptive expectationsAdaptive expectations

Partial adjustmentPartial adjustment

The The partial adjustment partial adjustment model describes themodel describes the

desired/optimal desired/optimal level of ylevel of ytt which is unobservable which is unobservable

adjustment equation adjustment equation looks as following wherelooks as following where denotes the fraction by which adjustment occursdenotes the fraction by which adjustment occurs

tttt wxy *

).y - )(y - (l y -y 1-t*t1-tt

Partial adjustmentPartial adjustment

If we solve the second equation for If we solve the second equation for yytt and insert the firstand insert the first

expression for expression for y*, y*, then we obtainthen we obtain::

This formulation offers a number of significant practicalThis formulation offers a number of significant practicaladvantages. It is intrinsically linear in the parametersadvantages. It is intrinsically linear in the parameters(unrestricted), (unrestricted), error termerror term nonautocorrelated nonautocorrelated therefore thereforethe parameters of this model can be estimatedthe parameters of this model can be estimatedconsistently and efficiently by ordinary least squaresconsistently and efficiently by ordinary least squares..

t1tt

t1-tttt

' w'' x' '

) - (l y )w - (1 )x - (l ) - (l y

ty

Partial adjustmentPartial adjustment

Consumer is viewed as a having desired level ofConsumer is viewed as a having desired level of

consumption, which is related to the current income.consumption, which is related to the current income.

When current income changes, inertial factors preventWhen current income changes, inertial factors prevent

An immediate movement to the new desired level ofAn immediate movement to the new desired level of

consumption. Instead, a partial movement is made, soconsumption. Instead, a partial movement is made, so

that: that:

with:with:

tdt INCCONS

ttdttt CONSCONSCONSCONS 11 1

10

Partial adjustmentPartial adjustment

This leads to an estimating form:This leads to an estimating form:

tttt INCCONSCONS 11 1

Partial adjustmentPartial adjustment

reg conspr l.conspr housedisp

Source | SS df MS Number of obs = 13-------------+------------------------------ F( 2, 10) = 1.19 Model | 12.1068721 2 6.05343604 Prob > F = 0.3447 Residual | 51.0017001 10 5.10017001 R-squared = 0.1918-------------+------------------------------ Adj R-squared = 0.0302 Total | 63.1085722 12 5.25904768 Root MSE = 2.2584

------------------------------------------------------------------------------conspr | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+----------------------------------------------------------------conspr | L1 | .3468319 .2890923 1.20 0.258 -.2973059 .9909698housedisp | .3928808 .3495421 1.12 0.287 -.3859475 1.171709_cons | 1.385804 2.130896 0.65 0.530 -3.362129 6.133738

Partial adjustment