Semi-Log Model. Semi-Log Model

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Semi-Log Model

n The slope coefficient measures the

relative change in Y for a given absolute change in the value of the explanatory variable (t).

Semi-Log Model

If we multiply the relative change in Y by 100, we get the percentage change or growth rate in

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Log (GDP) 1969-83

uGDP grew at the rate of .0269 per year, or at 2.69 percent per year.

uTake the antilog of 6.9636 to show that at the beginning of 1969 the estimated real GDP was about 1057 billions of dollars, i.e. at t = 0

Compound Rate of

Growth

n The slope coefficient measures the

instantaneous rate of growth

n How do we get r --the compound

growth rate? ub₂= ln (1 + r)

uantilog (b₂)= (1 + r)

uSo r = antilog (b₂) - 1

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Linear-Trend Model

n Trend model regresses Y on time.

uY_t = b₁ + b₂t+ e_t

uThis model shows whether GNP is increasing or decreasing over time

uThe model does not give the rate of growth.

uIf b₂ > 0, then an upward trend.

uIf b₂ < 0, thena downward trend.

Linear-Trend Example

n GNP = 1040.11 + 34.998t

n se (18.85) (2.07) R2 = .95

uGNP is increasing at the absolute amount of $35 billion per year.

uThere is a statistically significant upward trend.

n Growth model measures relative

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3. Lin-Log Models

Lin-Log Model

n The dependent variable is linear,

but the explanatory variable is in log form.

uUsed in situations for example where the rate of growth of the money supply affects GNP.

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Lin-Log Example

n GNP = b₁ + b₂lnM + e

uThe slope coefficient is dGNP/dlnM

FIt measures the absolute change in GNP for a relative change in M.

n If b₂ is 2000, a unit increase in the

log of the money supply increases GNP by $2000 billion.

uAlternatively, a 1% increase in the money supply increases GNP by 2000/100 = $20 billion.

Lin-Log Example

uIn this case, we need to divide by 100 since we are changing the money supply change from a relative change to a percentage change.

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4. Functional Form

Summary

Data

n GNP and money supply over the

period 1973-87 in the U.S.

n GNP in billions of dollars = Y

uMean = 2791.47

n M2 in billions of dollars = X

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Log-linear model

n lnY = 0.5531 + 0.9882lnX

n How interpret?

uThe slope coefficient is dlnY/dlnX

Fi.e. relative change in Y / relative change in X

uFor a 1% increase in the money supply, the average value of GNP increases by .9882% (almost 1%)

Log-linear model

uThe slope coefficient is also an elasticity:

uFor intercept: Y = antilog b1.

FThis is the average GNP when lnX = 0. Y * X X * Y X) (ln Y) (ln b₂ ∆ ∆ = ∆ ∆ =

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Log-Lin Model

n lnY = 6.8616 + 0.00057X

n How interpret?

uThe slope coefficient is dlnY/dX

Fi.e. relative change in Y / absolute change in X uFor a billion dollars rise in the

money supply, the log of GNP rises by .00057 per year.

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

Log-Lin Model

n How to convert to an elasticity?

uThe slope coefficient is:

GNP in increase 1.0007% a to leads supply money in the increase 1% a So 1.0007 5.67) .00057(175 X by multiply elasticity an get To X Y Y 1 X Y ln 2 =       ∂ ∂       = ∂ ∂ = b

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Lin-Log Model

n Y = -16329.0 + 2584.8lnX

n How interpret?

uThe slope coefficient is dY/dlnX

Fi.e. absolute change in Y/ relative change in X uA unit increase in the log of the

money supply increases GNP by 2584.8 billion dollars.

FIf money supply rises by 1%, GNP rises by $26 billion dollars.

Lin-Log Model

n How to convert to an elasticity?

uThe slope coefficient is:

GNP in increase .9260% a to leads supply money in the increase 1% A .9260 1.47 2584.8/279 Y by divide elasticity an get To X Y 1 X X ln Y 2 =       ∂ ∂       = ∂ ∂ = b

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Linear Model

n Y = 101.20 + 1.5323X

n How interpret?

uThe slope coefficient is dY/dX

Fi.e. absolute change in Y/ absolute change in X uFor a $1 billion increase in the

money supply increases GNP by $1.5323 billion dollars.

Linear Model

n How to convert to an elasticity?

uThe slope coefficient is: dY/dX

FMultiply this coefficient by Xbar/Ybar

• 1.5323 (1755.67/2791.47) = .9637

uA 1% increase in the money supply leads to a .9637% increase in GNP

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Monetarist Hypothesis

n Can test the monetarist hypothesis

with double log model

u1% increase in money supply leads to a 1% increase in GNP

FA t-test reveals that coefficient not different from 1.

Summary

n Models are similar:

uElasticities are similar.

uR2 _{are similar}

FCan only compare same similar dependent variables

uAll t values are significant

n Not much to choose among

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5. Reciprocal Model

Reciprocal Model

n Y = b₁ + b₂(1/X) + e

uModel is linear in the parameters, but nonlinear in the variables

uAs X increases,

FThe term 1/X approaches 0

FY approaches the limiting value of b₁.

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Fixed Cost Example

n Average fixed cost of production

declines continuously as output increases:

uFixed cost is spread over a larger and larger number of units and eventually becomes asymptotic.

Phillips Curve Example

n Sometimes the Philips curve is

expressed as a reciprocal model uY = b₁ + b₂(1/X) + e

FY = rate of change of money wages (inflation)

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Phillips Curve Example

natural unemployment rate than below.

FWages rise faster for a unit change in unemployment if the

unemployment rate is below the natural rate of unemployment than if it is above.

Phillips Curve Example

n Suppose we fit this model to data.

uUsing UK data 1950-66.

FY = -1.4282 + 8.7243 1/X Fse (2.068) (2.848)

uThis shows that the wage floor is -1.43%

FAs the unemployment rate increases indefinitely, the %

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6. Polynomial

Regression

Models

Polynomial Model

n These are models relating to cost

and production functions

n Ex: Long run average cost and

output

n LRAC curve is a U-shaped curve.

uCapture by a quadratic function (second degree polynomial):

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Polynomial Model

n In stochastic form:

uLRAC = b₁ + b₂Q + b₃Q2 _{+ e}

n We can estimate LRAC by OLS.

n Q and Q2 are correlated

uThey are not linearly correlated so do not violate the assumptions of CLRM.

S & L Example

n Use data for 86 S&Ls for 1975.

uOutput Q is measured as total assets

uLRAC is measured as average operating expenses as % of total assets

n Results:

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S & L Example

n This estimated function is

U-shaped.

n Its point of minimum average cost

if reached when total assets reach $569 billions: udLAC/dQ = -.615 + 2(.054) Q uSet equal to 0 F-.615 + .108 Q = 0 FQ = .615/.108 = 569

S & L Example

n This is used by regulators to

decide whether mergers are in the public interest and also by

managers to decide on efficient scale.

n It turns out that most S&Ls had

substantially less than $74 in assets, so mergers or growth ok.

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