Semi-Log Model - Πανεπιστήμιο Αιγαίου - Σχολή … Macpherson 3/22/99 Title...

David Macpherson 3/22/99

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Jump to first [email protected]

Semi-Log Modeln The slope coefficient measures the

relative change in Y for a givenabsolute change in the value of theexplanatory variable (t).


Semi-Log Modeln Using calculus:

in t change absolute

Yin change relative

tYY

t

Y

Y

1

t

Yln 2

=

∂

∂

=

∂∂

=

∂∂

=b

If we multiply the relative change in Y by 100,we get the percentage change or growth rate inY for an absolute change in t.


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Log (GDP) 1969-83n Log(Real GDP) = 6.9636 + 0.0269t

n se (.0151) (.0017)

n R2 = .95u GDP grew at the rate of .0269 per

year, or at 2.69 percent per year.u Take the antilog of 6.9636 to show

that at the beginning of 1969 theestimated real GDP was about1057 billions of dollars, i.e. at t = 0


Compound Rate ofGrowthn The slope coefficient measures the

instantaneous rate of growth

n How do we get r --the compoundgrowth rate?u b2= ln (1 + r)u antilog (b2)= (1 + r)u So r = antilog (b2) - 1u So r = antilog(.0269) -1

F r= 1.0273 - 1 = .0273


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Linear-Trend Modeln Trend model regresses Y on time.

u Yt = b1 + b2t+ et

u This model shows whether GNP isincreasing or decreasing over time

u The model does not give the rateof growth.

u If b2 > 0, then an upward trend.u If b2 < 0 , then a downward trend.


Linear-Trend Examplen GNP = 1040.11 + 34.998t

n se (18.85) (2.07) R2 = .95u GNP is increasing at the absolute

amount of $35 billion per year.u There is a statistically significant

upward trend.

n Growth model measures relativeperformance

n Trend model measures absoluteperformance


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3. Lin-Log Models


Lin-Log Modeln The dependent variable is linear,

but the explanatory variable is inlog form.u Used in situations for example

where the rate of growth of themoney supply affects GNP.


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Lin-Log Examplen GNP = b1 + b2lnM + e

u The slope coefficient is dGNP/dlnMF It measures the absolute change in

GNP for a relative change in M.

n If b2 is 2000, a unit increase in thelog of the money supply increasesGNP by $2000 billion.u Alternatively, a 1% increase in the

money supply increases GNP by2000/100 = $20 billion.


Lin-Log Exampleu In this case, we need to divide by

100 since we are changing themoney supply change from arelative change to a percentagechange.


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4. Functional FormSummary


Datan GNP and money supply over the

period 1973-87 in the U.S.

n GNP in billions of dollars = Yu Mean = 2791.47

n M2 in billions of dollars = Xu Mean = 1755.67


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Log-linear modeln lnY = 0.5531 + 0.9882lnX

n How interpret?u The slope coefficient is dlnY/dlnX

F i.e. relative change in Y /relative change in X

u For a 1% increase in the moneysupply, the average value of GNPincreases by .9882% (almost 1%)


Log-linear modelu The slope coefficient is also an

elasticity:

u For intercept: Y = antilog b1.F This is the average GNP when lnX

= 0.

Y*X

X*Y

X)(ln

Y) (lnb2 ∆

∆=

∆∆

=


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Log-Lin Modeln lnY = 6.8616 + 0.00057X

n How interpret?u The slope coefficient is dlnY/dX

F i.e. relative change in Y / absolute change in X

u For a billion dollars rise in themoney supply, the log of GNPrises by .00057 per year.

F To make a %, multiply by 100:GNP rises by 0.057% per year.


Log-Lin Modeln How to convert to an elasticity?

u The slope coefficient is:

GNPin increase 1.0007% a toleads

supply money in the increase 1% a So

1.0007 5.67).00057(175

Xby multiply elasticityan get To

X

Y

Y

1

X

Yln 2

=

∂∂

=

∂∂

=b


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Lin-Log Modeln Y = -16329.0 + 2584.8lnX

n How interpret?u The slope coefficient is dY/dlnX

F i.e. absolute change in Y/ relative change in X

u A unit increase in the log of themoney supply increases GNP by2584.8 billion dollars.

F If money supply rises by 1%, GNPrises by $26 billion dollars.


Lin-Log Modeln How to convert to an elasticity?

u The slope coefficient is:

GNPin increase .9260% a toleads

supply money in the increase 1%A

.9260 1.472584.8/279

Yby divide elasticityan get To

X

Y

1

X

Xln

Y2

=

∂∂

=

∂∂

=b


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Linear Modeln Y = 101.20 + 1.5323X

n How interpret?u The slope coefficient is dY/dX

F i.e. absolute change in Y/absolute change in X

u For a $1 billion increase in themoney supply increases GNP by$1.5323 billion dollars.


Linear Modeln How to convert to an elasticity?

u The slope coefficient is: dY/dXF Multiply this coefficient by

Xbar/Ybar• 1.5323 (1755.67/2791.47) = .9637

u A 1% increase in the money supplyleads to a .9637% increase in GNP


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Monetarist Hypothesisn Can test the monetarist hypothesis

with double log modelu 1% increase in money supply

leads to a 1% increase in GNPF A t-test reveals that coefficient not

different from 1.


Summaryn Models are similar:

u Elasticities are similar.

u R2 are similarF Can only compare same similar

dependent variables

u All t values are significant

n Not much to choose amongmodels.u Depends on issue-elasticity,

growth, absolute change, etc.


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5. Reciprocal Model


Reciprocal Modeln Y = b1 + b2(1/X) + e

u Model is linear in the parameters,but nonlinear in the variables

u As X increases,F The term 1/X approaches 0F Y approaches the limiting value of

b1.


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Fixed Cost Examplen Average fixed cost of production

declines continuously as outputincreases:u Fixed cost is spread over a larger

and larger number of units andeventually becomes asymptotic.


Phillips Curve Examplen Sometimes the Philips curve is

expressed as a reciprocal modelu Y = b1 + b2(1/X) + e

F Y = rate of change of moneywages (inflation)

F X = unemployment rate.


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Phillips Curve Exampleu The curve is steeper above the

natural unemployment rate thanbelow.

F Wages rise faster for a unit changein unemployment if theunemployment rate is below thenatural rate of unemployment thanif it is above.


Phillips Curve Examplen Suppose we fit this model to data.

u Using UK data 1950-66.F Y = -1.4282 + 8.7243 1/XF se (2.068) (2.848)

u This shows that the wage floor is -1.43%

F As the unemployment rateincreases indefinitely, the %decrease in wages will not be morethan 1.43 percent per year.


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6. Polynomial Regression Models


Polynomial Modeln These are models relating to cost

and production functions

n Ex: Long run average cost andoutput

n LRAC curve is a U-shaped curve.u Capture by a quadratic function

(second degree polynomial):u LRAC = b1 + b2Q + b3Q

2


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Polynomial Modeln In stochastic form:

u LRAC = b1 + b2Q + b3Q2 + e

n We can estimate LRAC by OLS.

n Q and Q2 are correlatedu They are not linearly correlated so

do not violate the assumptions ofCLRM.


S & L Examplen Use data for 86 S&Ls for 1975.

u Output Q is measured as totalassets

u LRAC is measured as averageoperating expenses as % of totalassets

n Results:u LRAC = 2.38 -.615Q + .054 Q2


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S & L Examplen This estimated function is U-

shaped.

n Its point of minimum average costif reached when total assets reach$569 billions:u dLAC/dQ = -.615 + 2(.054) Qu Set equal to 0

F -.615 + .108 Q = 0F Q = .615/.108 = 569


S & L Examplen This is used by regulators to

decide whether mergers are in thepublic interest and also bymanagers to decide on efficientscale.

n It turns out that most S&Ls hadsubstantially less than $74 inassets, so mergers or growth ok.


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END OFCHAPTER 6

Semi-Log Model - Πανεπιστήμιο Αιγαίου - Σχολή … Macpherson 3/22/99 Title...

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