Functional Forms

8/6/2019 Functional Forms

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QE 8,1

8 FUNCTIONAL FORMS

All models-considered thus far linear in both parameters &

variables.

1 2 3

2

1 2 3 1 2 3

2

3

-linear in parameters (LIP) & linear invariables (LIV)

- linear in parameters ( , and ),

but not linear-all-variables [i.e. (e.g. in , appears-power- ' 2 ')]

whi

Y b b x b x

Y b b x b x b b b

xs b x x

= + +

= + +

2 2

1 2 2

lst -not linear in all - parameters since enters -power- 'Y b b x b= +

But-many economic phenomena-relationship between- variables

not linear e.g. if -want-calculate elasticity values good, - slope

coefficient gives - absolute changes - one variable given a unit

change in the other.

Hence,-using-alternative functional form, - can still use OLS

-calculate these elasticities.

But - use OLS, models must - linear - parameters, but not

necessarily in their variables.

Although -several models, we-consider:

Log-linear model

Semilog models

Polynomial regression models

Regression through the Origin


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QE 8,2

8.1 THE LOG-LINEAR/ LOG-LOG/ DOUBLE LOG

MODEL

(a) The Two-variable Model

2B

ii AXY = - non-linear in variables.

but, taking logarithms,

22lnlnln XBAY

i+=

This can be estimated as

iiiuXBBY ++=

*

221

*

*

1

*

2 2

where ln

ln ,

ln and

is - disturbance term

i i

i i

i

Y Y

B A

X X

u

=

=

=

Model-now linear in parameters (and also in the transformed

variables Y* andX*).

regression can be estimated with OLS and -estimators - BLUE,

provided - usual assumptions hold - transformed model.


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QE 8,3

*

22 *

22 2 2

2

ln

ln

i i i i

i ii i i

i

YY Y Y X Y

BX Y X X X

X

= = = =

i.e.,B2 measures the elasticity ofYwith respect toX2, and thus -

can - interpreted - the %age change in Y for a given %age

change inX.

Thus-in fig. (b) -slope- gives-estimate-price elasticity and since

it- straight line, the elasticity-constant throughout: known - constant

elasticity model(use this model only where elasticity - expected -

constant).

Example:

Weekly lotto expenditure (Y) in relation to weekly personaldisposable income (X) ($).


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QE 8,4

The OLS regression based-data above give:

LnYi = -0.672 + 0.7256 lnXi

p = (0.2676) (0.0001) r2 = 0.8644

and -results - interpreted as ff:

the expenditure elasticity is 0.73 i.e. if PDI increases by 1%

expenditure on Lotto on the average increases 0.73%

(ep


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QE 8,5

Example: Cobb-Douglas production function:

KALY =

where L is total labour input

Kis total capital input

A, and are parameters.

Then, taking logs,

KlnLlnAlnYln

++=

.

We can model this as:

i

*

i3

*

i21

*

i uKBLBBY +++=

te red i s t urbancaisu

an dKlnK

LlnL

YlnY

B

B

e r c e p t in tBw h e r e

i

i

*

i

i

*

i

i*i

3

2

1

=

=

=

=

=

=


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OLS regression based-data above give:


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Ln Yi = -1.6524 + 0.3397lnLi + 0.8460lnKi

p = (0.014) (0.085) (0.000)

r2 = 0.995 F = 1719.23 (0.000)

and results can be interpreted as follows:

Holding capital input constant, if labour input increases by

1%, on the average, output increases by 0.34%.

Holding labour input constant, if capital input increases by

1%, on the average, output increases by 0.85%.

Estimated coefficients: labour is individually statistically

significant at 10% level whilst capital is (individually)

statistically significant at all levels.

The r2 value of 0.995 is that 99.5% of the variation in the log

of output is explained by the variation in the logs of capital

and labour.

Estimated F value-so highly significant that can reject null

hypothesis that labour and capital together have no impact on

output

Adding the two elasticity coefficients gives -economic

parameter- returns to scale parameter i.e. response of output

to a proportional change in inputs.

Our example- these sum to 1.1857-indicating-increasing

returns to scale (why?).


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8.2 COMPARING LINEAR AND LOG-

LINEAR MODELS

Economic theory does not always specify - particular

functional form of relationship between variables.

How-choose between competing models?

Plot the data: if scattergram shows relationship-

linear then linear specification might appropriate and if

shows -non-linear relationship then log-linear model-suitable.

This principle-however works only simple case of

two variable regression model, but for multiple regressions

other guidelines -needed.

What about choosing models basis of

comparing r2

i.e. choose model gives highest r2

?

This approach-own problems?

To compare r2values two models, the

dependent variable must-same form. And if different, then -

not directly comparable.

Even if-dependent variables both models

same still need careful since r2 can always-increased adding

more explanatory variables.

Hence instead -focussing mainly on r2 ,

need-consider factors such as :


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Relevance of variables included model.

Expected signs of coefficients.

Their statistical significance.

And other derived measures like elasticity

coefficients.

8.3 THE SEMILOG MODELS

8.3.1 The log-lin (Growth) Model

Often used to measure growth rates.

Consider GDP, Y. The growth rate can be modelled as

follows:

0 (1 )

t

t

Y Y r= +

where r is the compound growth rate

Then:

0ln ln ln(1 )

tY Y t r = + +

This can be modelled as:

)r1ln(B

andYlnBwhereutBBYln

1

00

t10t

+=

=

++=

The above is called a semilog model because only one

variable (in this case the dependent) appears in logarithmic form

called LOG-LIN model.

Example:Population of United States (millions of people), 1970-1999.


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Ln(USpop)Yi = 5.3170 + 0.0098tp = (0.0000) (0.0000) r2 = 0.9996

and -results can be interpreted as follows:

the slope coefficient of 0.0098 means on the average the

logof Y (US population) has been increasing at the rate of

0.0098 per year or alternatively, that Y has been increasing at

the rate of 0.98% per year.

i.e. in a log-lin model the slope coefficient measures the

proportional or relative change in Yfor a given absolute

change in the explanatory variable, time, in our example.


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If this relative change is multiplied by 100, -obtain %age

change orgrowth rate.

8.3.2 The lin-log Model

previous section- considered growth model, - dependent

variable was log form but explanatory variable was linear form.

If - dependent variable - linear but - explanatory variable(s)

is/are logarithmic, called LIN-LOG model.

e.g. we want to find out how expenditure on services (Y) behaves

if total personal consumption expenditure (X) increases by a

certain percentage.

i.e.1 2

lni i i

Y X u = + +

Thus, 2measures the absolute change in Yif the log ofX

changes by one unit.

Example: Quarterly expenditure on services (Y) and total personal

expenditure (X) 1993-11998-3.


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2431.69lnX-17907.5Y +=

if -log ofXchanges by one unit, - absolute change in Y

will be 2431.69 billions.


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And since a change in the log of a number is a relative

change, to calculate the absolute change in Y for a 1%

change in X divide the estimated slope coefficient by

100 (i.e. 2100

).

In-example: ifXchanges by 1 %, on the average, Ywill

change by 24.31 billions.

There is no reason why you cannot have more complex

models with more than one log term or why you cannot combine

log and linear terms as explanatory variables.

8.4 POLYNOMIAL REGRESSION MODELS

Consider the model:

2 3

0 1 2 3i i i i i Y X X X u = + + + +


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These models used extensively in applied econometric

studies relating to production and cost functions.

Example:

These polynomial models can be evaluated readily by OLS,

since even though the variables are perfectly correlated, the

correlation is not linear.


= + +2 3 141.77 63.48 - 12.96 0.94i i i i Y X X X

8.5 REGRESSION THROUGH THE ORIGIN

Yi = 2Xi + ui

In this model the intercept is absent or zero.


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If this is the case, the formulae forb2, its variance, and the

regression variance are modified as shown on pp. 274 of Gujarati

(the modifications are obvious).

However, note the following:

eineed not be zero.

R2 can lie outside the range 0-1.

This model should not be used unless there are strong a

priori reasons for doing so i.e. it is only appropriate if theorystipulates there should be no intercept.

Functional Forms

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