Ch.6 Simple Linear Regression: Continued

1
Ch.6 Simple Linear Regression Continued

• To complete the analysis of the simple linear
  regression model, in this chapter we will
  consider
• how to measure the variation in yt, that is
  explained by the model
• how to report the results of a regression
  analysis
• how changes in the units of measurement affect
  the estimates
• some alternative functional forms that may be
  used to represent possible relationships between
  yt and xt.

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The Coefficient of Determination (R2)

• Two major reasons for analyzing the model
• y ?1 ?2x e
• are
• To explain how the dependent varaible (yt)
  changes as the independent variable (xt) changes
• To predict yo given xo.
• We want the independent variable (xt) to explain
  as much of the variation in the dependent
  variable (yt) as possible.
• We introduced the independent variable (xt) in
  hope that its variation will explain the
  variation in y
• A measure of goodness of fit will measure how
  much of the variation in the dependent variable
  (yt) has been explained by variation in the
  independent variable (xt).

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Separate yt into its explainable and
unexplainable components
is explainable.
where
The error term et is unexplainable. Using our
estimates for ?1 and ?2, we get estimates of
E(yt) and our residuals give us estimates of the
error terms.
Residual is defined as the difference between
the actual and the predicted values of y.
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The total variation in yt is measured as the sum
of the squared deviations from the mean
Also known as SST (Total Sum of Squares)
A single deviation of yt from its mean can be
split into two parts
The sum of squared deviations from the mean is
This term is zero
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Graphically, a single y deviation from mean can
be split into the two parts
Unexplained
Total Variation
?
Explained
xt
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Analysis of Variance (ANOVA)
SST SSR SSE

• Where
• SST Total Sum of Squares with T-1 degrees of
  freedom. It measures the total variation in the
  actual yt values about its mean.
• SSR Regression Sum of Squares with 1 degree of
  freedom. It measures the variation in the
  predicted values of yt about their mean. It is
  the part of the total variation that is explained
  by the model.
• SSE Error Sum of Squares with T-2 degrees of
  freedom. It measures the variation in the actual
  yt values about the predicted yt values. It is
  the part of the total variation that is left
  unexplained.

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R2 SSR/SST 1 SSE/SST
SSR
SSE
SST
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Coefficient of Determination R2

• R2 is the proportion of the total variation (SST)
  that is explained by the model. We can also think
  of it as one minus the proportion of the total
  variation that is unexplained (left in the
  residuals).
• 0 ? R2 ? 1
• The closer R2 is to 1.0, the better the fit of
  the model and the greater is the predictive
  ability of the model over the sample.
• If R2 1 ? the model has explained everything.
  All the data points lie on the regression lie
  (very unlikely). There are no residuals.
• If R2 0 ? the model has explained nothing.

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Graph A
y
R2 appears to be 1.0. All data Points lie on a
line.
x
Graph B
y
R2 appears to be 0. The best line thru
these points appears to have a slope of zero.
x
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Graph C
y
R2 appears to be close to 1.0.
x
Graph D
y
R2 appears to be greater than 0 but less than R2
in graph C.
x
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• In the food expenditure example, R2 0.317 ?
  31.7 of the total variation in food
  expenditures has been explained by variation in
  household income.
• More Examples

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Correlation Analysis

• Correlation coefficient between x and y is
• The Sample Correlation between x and y is
• It is always true that
• -1 ? r ? 1
• It measures the strength of a linear relationship
  between x and y.

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Correlation and R2

• It can be shown that the square of the sample
  correlation coefficient for x and y is equal to
  R2.
• R2 can also be computed as the square of the
  sample correlation coefficient for the y values
  and the values.
• It can also be shown that

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Reporting Regression Results
(s.e.) (22.139) (0.0305) R2
0.317

• The numbers in parentheses are the standard
  errors of the coefficients estimates. These can
  be used to construct the necessary t-statistics
  to ascertain the significance of the estimates.
• Sometimes, authors will report the t-statistic
  instead of the standard error. This would be the
  t-statistic for the Ho ? 0

(t-stat) (1.841) (4.201) R2
0.317
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Units of Measurement

• b1 is measured in y units
• b2 is measured in y units over x units
• Example 3.15 from Chapter 3 Exercises
• y number of sodas sold x
  temperature in degrees (oF)

If xo 0o then the model predicts So b1 is
measured in y units ( of sodas). b2 6 where 6
is in ( of sodas / degrees). If x increases by
10 degrees ? y increases by 60 sodas

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• Let newx x/100.
• no change to b1
• b2 increases by 100

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Functional Forms

• A linear model is one that is linear in the
  parameters with an additive error term.
• y ?1 ?2x e
• The coefficient ?2 measures the effect of a one
  unit change in x on y. As the model is written
  above, this effect is assumed to be constant
• However, we want to have the ability to model
  relationships among economic variables where the
  effect of x on y is not constant.
• Example our food expenditure example assumes
  that the increase in food spending from an
  additional dollar of income was the same whether
  the family had a high or low income. We can
  capture these effects using logs, powers and
  reciprocals yet still maintain a model that is
  linear in the parameters with an additive error
  term.

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The Natural Logarithm

• We will use the derivative property often
• Let y be the log of X
• y ln(x) ? dy/dx 1/x or dy dx/x
• This means that the absolute change in the log of
  X is equivalent to the relative change in the
  level of X.
• Let x50 ? ln(x) 3.912
• Let x52 ? ln(x) 3.951
• ? dln(x) 3.951 3.912 0.039
• The absolute change in ln(x) is 0.039, which can
  be interpreted as a relative change in X (X
  increases from 50 to 52, which, in relative
  terms, is 3.9)

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Using Logs
What does ?2 measure?
What does ?2 measure?
What does ?2 measure?
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Example Y food , X Weekly Income
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Log- Linear Model
Double Log Model