Class Outline

1
Class Outline

• Functional Forms of Regression Models
• Regression Through the Origin
• Log Linear Model
• Comparing Linear and Log Linear Models
• Multiple Log-Linear Models
• The Semilog Model
• The Lin-Log Model
• Reciprocal Models
• Polynomial Regression Models
• Summary of Functional Forms
• Reading Chapter 6

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Regression Through the Origin

• The intercept is absent or zero. It can be shown
  that

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Regression Through the Origin

• Characteristics of this model
• In calculating the parameter we use raw and cross
  products instead of mean-adjusted sums of squares
  and cross products
• The degrees of freedom are n-1 rather than n-2
• The formula for r2 includes an intercept.
  Therefore you should not use this formula, or you
  will obtain nonsensical results, like negative r2
• The sum of the residuals is always zero, but in
  the case of a model with intercept this could not
  be the case.

4
Log-Linear Model

• Remember our example with the Lotto regression.
• Assume that our expenditure function is as
  follows
• Where Y is the expenditure on Lotto and X is
  personal disposable income. The model is
  nonlinear in the variable X

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Log-Linear Model

• We can transform the equation in logarithms
• For estimation purposes we can write this model
  as

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Log-Linear Model

• This is a linear regression model for the
  parameters ?1 and ?2.
• Observe that the slope coefficient ?2 measures
  the elasticity of Y with respect to X, that is,
  the percentage change in Y for a given percentage
  change in X.

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Log-Linear Model

• We can define the elasticity as

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Log-Linear Model
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Comparing Linear and Log-Linear Models

• Assume we run a linear model and a log linear
  model for the same dataset, which one to choose?
• Plot the data and see if you can determine the
  functional form
• We cannot compare r2 of log linear and linear
  models. By definition in the linear model r2
  measures the proportion of the variation in Y
  explained X, while in the log linear model r2
  explains the proportion of the variation of lnY
  explained by lnX. These measures are different

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Comparing Linear and Log-Linear Models

• The variation in Y is a absolute change, while
  the variation in log of Y measures the relative
  or proportional change.
• Even if the dependent variables of two models is
  the same, we should not choose our models based
  on the highest r2 criterion. This is because this
  measure changes with the addition of variables.

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Multiple Log-Linear Regression Models

• Example The Cobb Douglas Production Function

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How to Measure the Growth Rate The Semilog Model

• When we are interested on the growth rate of some
  economic variables we can use this model.
• Example we want to measure the growth rate of
  population over the period 1970-1999
• Y0 beginning value of Y
• Yt Ys value at time t
• r the compound rate of growth over time

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How to Measure the Growth Rate The Semilog Model

• Lets transform this equation as follows

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The Lin-Log Model When the Explanatory Variable
is Logarithmic

• In this case, the independent variable, X, is
  expressed in logarithm
• Example we want to find how expenditure on
  services (Y) behaves if total personal
  consumption expenditure (X) increases

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The Lin-Log Model When the Explanatory Variable
is Logarithmic

• This equation states that the absolute change in
  Y is equal to ?2 times the relative change in X.

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Reciprocal Models

• These models are used when the functional form
  have some asymptotic characteristics

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Reciprocal Models
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Polynomial Regression Models

• We can estimate this model with OLS. The only
  problem is the presence of multicollinearity, but
  usually this problem should not be too important
  in this case because the explanatory variables
  are not linear functions of X.

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Polynomial Regression Models
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Summary of Functional Forms