Chapter 6: Form

1
Chapter 6 Form Specification
2

• 1. Log-Linear Models

3
Functional Form

• OLS applies to models that are linear in the
  parameters.
• Models do not have to be linear in the variables.
  
• Up to now we have dealt with models linear in the
  variables.
• Now look models are linear in the parameters, but
  not in the variables.

4
Cobb-Douglas Production Function

• Y AK?L?
• Y output
• K capital input
• L labor input
• This is a nonlinear relationship, but we can
  transform it
• lnK 2 is 2lnK
• ln(KL) is lnK lnL

5
Cobb-Douglas Production Function

• So lnY ln A ?lnK ?lnL
• This is linear in the parameters but nonlinear in
  the variables
• It is linear in the logs of the variables
• It is called a log-log, double-log or log-linear
  model.

6
Cobb-Douglas Production Function

• Estimating equation
• lnY b1 b2lnK b3lnL e
• So now it is a linear model
• linear in the transformed variables so we can use
  OLS and get BLUE.

7
Elasticity

• The slope coefficient of a log-linear model
  measures the elasticity of Y with respect to X.

So the coefficient is an elasticity.
8
Elasticity

• This elasticity is constant over XY
• The slope is not constant.
• The slope coefficient varies according to X and
  Y.
• In linear model, the slope is constant but the
  elasticity varies.
• To get the elasticity, multiply the slope
  coefficient by X/Y.
• Typically we do this at the means of X and Y

9
Interpretation

• b2 is the partial elasticity of output with
  respect to the capital input, holding labor
  constant.
• It measures the change in output for a given
  change in capital, holding labor constant.
• b3 is the partial elasticity of output with
  respect to the labor input, holding capital
  constant.

10
Returns to Scale

• b2 b3 measures returns to scale
• The response of output to a proportionate change
  in inputs.
• If b2 b3 1 constant returns
• Doubling inputs doubles outputs.
• If b2 b3 lt1 decreasing returns
• If b2 b3 gt increasing returns

11
Cobb-Douglas Example

• 1958-72 agricultural Taiwan data
• lnY -3.34 0.49 lnK 1.50 lnL
• t (-1.36) (4.80) (0.54)
• R2 .89
• Y is GNP in millions of
• K is real capital in millions of
• L is millions of man days

12
Cobb-Douglas Example

• Output elasticity of capital is 0.49
• Holding labor constant, a 1 increase in the
  capital input leads to a 0.49 increase in
  output.
• Output elasticity of labor is1.50
• Holding capital constant, a 1 increase in the
  labor input leads to a 1.5 increase in output.

13
Cobb-Douglas Example

• Increasing returns to scale since b2 b3
  1.99.
• Intercept (-3.34) is average value of ln Y when
  lnK and lnL are zero
• Take antilog to get Y.04 million
• R2 means that 89 of the variation in the log of
  output is explained by the logs of labor and
  capital.

14
Linear vs Log Linear

• Cant choose model with highest R2 since our
  dependent variable is not the same
• Y in one and ln Y in another.
• Variation in Y is an absolute change, whereas
  variation in log Y is a proportional change.
• Better to choose on basis of signs of
  coefficients, elasticities and if theory makes
  sense.

15
Demand Function

• We can model demand as a log linear model and so
  estimate the elasticity of demand
• lnQ b1 b2 ln Pown b3ln Pother
• Q is coffee consumption per day
• Pown is price of coffee per pound
• Pother is price of tea per pound

16
Demand Function Results

• Results
• lnQ0.78 -0.25lnPown 0.38lnPother
• t (51.1) (-5.12) (3.25)
• Own price elasticity is -.25.
• Holding other things constant, if the price
  increases by 1 the quantity demanded will
  decrease by 0.25.
• This is inelastic - less than 1 in absolute
  value.

17
Demand Function Results

• Cross-price elasticity is .38.
• Holding other things constant, if the price of
  tea increases by 1, the quantity of coffee
  demanded increases by 0.38
• If cross-price elasticity is positive, they are
  substitute products.
• If cross-price elasticity is negative, then
  complementary products.

18
Demand Function With Income

• Housing demand model
• ln Q 4.17 -.25 ln P .96 ln Y se
  (.11) (.017) (.026)
• Price elasticity is -.25 (inelastic)
• Income elasticity ( change in Q/ change in
  income) is .96

19
Demand Function With Income

• Can test whether income elasticity is
  significantly different from 1
• Ho b3 1
• t0.96-1/.026 -1.54
• Critical t value at 5 is 1.96
• We cannot reject the hypothesis that the income
  elasticity of demand is 1.

20

• 2. Semi-Log Models

21
Semi-Log Model

• Use this model when we are interested in the rate
  of growth of certain variables such as GNP or
  employment.

22
Compound Interest

• Compound interest formula
• Yt Yo (1 r)t
• Yo is the initial value of Y
• Yt is the value of Y at time t
• r is the compound rate of growth of Y

23
Compound Interest Example

• Suppose we have 1000 and we put it in the bank
  at 5 interest. After one year it will have
  grown to 1000(1.05) 1050
• After two years it will have grown to
  1050(1.05) 1102.5
• This is the same as 1000(1.05)2 1000 (1.1025)
  1102.5

24
Compound Interest

• Take natural log of compound interest formula
• ln Yt ln Yo t ln(1 r)
• Let B1 ln Yo
• Let B2 ln (1 r)
• Rewrite and add error term
• ln Yt B1 B2 t e
• This is a semilog model since only one variable
  is in logarithmic form.

25

26
Semi-Log Model

• The slope coefficient measures the relative
  change in Y for a given absolute change in the
  value of the explanatory variable (t).

27
Semi-Log Model

• Using calculus

If we multiply the relative change in Y by 100,
we get the percentage change or growth rate in Y
for an absolute change in t.
28
Log (GDP) 1969-83

• Log(Real GDP) 6.9636 0.0269t
• se (.0151) (.0017)
• R2 .95
• GDP grew at the rate of .0269 per year, or at
  2.69 percent per year.
• Take 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

29
Compound Rate of Growth

• The slope coefficient measures the instantaneous
  rate of growth
• How do we get r --the compound growth rate?
• b2 ln (1 r)
• antilog (b2) (1 r)
• So r antilog (b2) - 1
• So r antilog(.0269) -1
• r 1.0273 - 1 .0273

30
Linear-Trend Model

• Trend model regresses Y on time.
• Yt b1 b2t et
• This model shows whether GNP is increasing or
  decreasing over time
• The model does not give the rate of growth.
• If b2 gt 0, then an upward trend.
• If b2 lt 0 , then a downward trend.

31
Linear-Trend Example

• GNP 1040.11 34.998t
• se (18.85) (2.07) R2 .95
• GNP is increasing at the absolute amount of 35
  billion per year.
• There is a statistically significant upward
  trend.
• Growth model measures relative performance
• Trend model measures absolute performance

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• 3. Lin-Log Models

33
Lin-Log Model

• The dependent variable is linear, but the
  explanatory variable is in log form.
• Used in situations for example where the rate of
  growth of the money supply affects GNP.

34
Lin-Log Example

• GNP b1 b2lnM e
• The slope coefficient is dGNP/dlnM
• It measures the absolute change in GNP for a
  relative change in M.
• If b2 is 2000, a unit increase in the log of the
  money supply increases GNP by 2000 billion.
• Alternatively, a 1 increase in the money supply
  increases GNP by 2000/100 20 billion.

35
Lin-Log Example

• In 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

37
Data

• GNP and money supply over the period 1973-87 in
  the U.S.
• GNP in billions of dollars Y
• Mean 2791.47
• M2 in billions of dollars X
• Mean 1755.67

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Log-linear model

• lnY 0.5531 0.9882lnX
• How interpret?
• The slope coefficient is dlnY/dlnX
• i.e. relative change in Y /
  relative change in X
• For a 1 increase in the money supply, the
  average value of GNP increases by .9882 (almost
  1)

39
Log-linear model

• The slope coefficient is also an elasticity
• For intercept Y antilog b1.
• This is the average GNP when lnX 0.

40
Log-Lin Model

• lnY 6.8616 0.00057X
• How interpret?
• The slope coefficient is dlnY/dX
• i.e. relative change in Y / absolute
  change in X
• For a billion dollars rise in the money supply,
  the log of GNP rises by .00057 per year.
• To make a , multiply by 100 GNP rises by 0.057
  per year.

41
Log-Lin Model

• How to convert to an elasticity?
• The slope coefficient is

42
Lin-Log Model

• Y -16329.0 2584.8lnX
• How interpret?
• The slope coefficient is dY/dlnX
• i.e. absolute change in Y/ relative change
  in X
• A unit increase in the log of the money supply
  increases GNP by 2584.8 billion dollars.
• If money supply rises by 1, GNP rises by 26
  billion dollars.

43
Lin-Log Model

• How to convert to an elasticity?
• The slope coefficient is

44
Linear Model

• Y 101.20 1.5323X
• How interpret?
• The slope coefficient is dY/dX
• i.e. absolute change in Y/ absolute change in X
• For a 1 billion increase in the money supply
  increases GNP by 1.5323 billion dollars.

45
Linear Model

• How to convert to an elasticity?
• The slope coefficient is dY/dX
• Multiply this coefficient by Xbar/Ybar
• 1.5323 (1755.67/2791.47) .9637
• A 1 increase in the money supply leads to a
  .9637 increase in GNP

46
Monetarist Hypothesis

• Can test the monetarist hypothesis with double
  log model
• 1 increase in money supply leads to a 1
  increase in GNP
• A t-test reveals that coefficient not different
  from 1.

47
Summary

• Models are similar
• Elasticities are similar.
• R2 are similar
• Can only compare same similar dependent variables
• All t values are significant
• Not much to choose among models.
• Depends on issue-elasticity, growth, absolute
  change, etc.

48

• 5. Reciprocal Model

49
Reciprocal Model

• Y b1 b2(1/X) e
• Model is linear in the parameters, but nonlinear
  in the variables
• As X increases,
• The term 1/X approaches 0
• Y approaches the limiting value of b1.

50
Fixed Cost Example

• Average fixed cost of production declines
  continuously as output increases
• Fixed cost is spread over a larger and larger
  number of units and eventually becomes
  asymptotic.

51
Phillips Curve Example

• Sometimes the Philips curve is expressed as a
  reciprocal model
• Y b1 b2(1/X) e
• Y rate of change of money wages (inflation)
• X unemployment rate.

52
Phillips Curve Example

• The curve is steeper above the natural
  unemployment rate than below.
• Wages rise faster for a unit change in
  unemployment if the unemployment rate is below
  the natural rate of unemployment than if it is
  above.

53
Phillips Curve Example

• Suppose we fit this model to data.
• Using UK data 1950-66.
• Y -1.4282 8.7243 1/X
• se (2.068) (2.848)
• This shows that the wage floor is -1.43
• As the unemployment rate increases indefinitely,
  the decrease in wages will not be more than
  1.43 percent per year.

54

• 6. Polynomial
• Regression Models

55
Polynomial Model

• These are models relating to cost and production
  functions
• Ex Long run average cost and output
• LRAC curve is a U-shaped curve.
• Capture by a quadratic function (second degree
  polynomial)
• LRAC b1 b2Q b3Q2

56
Polynomial Model

• In stochastic form
• LRAC b1 b2Q b3Q2 e
• We can estimate LRAC by OLS.
• Q and Q2 are correlated
• They are not linearly correlated so do not
  violate the assumptions of CLRM.

57
S L Example

• Use data for 86 SLs for 1975.
• Output Q is measured as total assets
• LRAC is measured as average operating expenses as
  of total assets
• Results
• LRAC 2.38 -.615Q .054 Q2

58
S L Example

• This estimated function is U-shaped.
• Its point of minimum average cost if reached when
  total assets reach 569 billions
• dLAC/dQ -.615 2(.054) Q
• Set equal to 0
• -.615 .108 Q 0
• Q .615/.108 569

59
S L Example

• This is used by regulators to decide whether
  mergers are in the public interest and also by
  managers to decide on efficient scale.
• It turns out that most SLs had substantially
  less than 74 in assets, so mergers or growth ok.

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• END OF CHAPTER 6