Ch.6 Simple Linear Regression: Continued - PowerPoint PPT Presentation

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Ch.6 Simple Linear Regression: Continued

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Title: 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.

2
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).

3
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.
4
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
5
Graphically, a single y deviation from mean can
be split into the two parts
Unexplained
Total Variation
?
Explained
xt
6
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.

7
R2 SSR/SST 1 SSE/SST
SSR
SSE
SST
8
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.

9
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
10
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
11
  • 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

12
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.

13
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

14
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
15
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

16
  • Let newx x/100.
  • no change to b1
  • b2 increases by 100

17
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.

18
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)

19
Using Logs
What does ?2 measure?
What does ?2 measure?
What does ?2 measure?
20
Example Y food , X Weekly Income
21
Log- Linear Model
Double Log Model
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