PUAF 610 TA

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PUAF 610 TA

• Session 10

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TODAY

• Ideas about Final Review
• Regression Review

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Final Review

• Any idea about the final review next week?
• Go over lectures
• Go over problem sets that related to the exam
• Go over extra exercises
• Try to get information from instructors
• Email me your preferences

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Regression

• In regression analysis we analyze the
  relationship between two or more variables.
• The relationship between two or more variables
  could be linear or non linear.
• Simple Linear Regression y, x
• Multiple Regression y, x1, x2, x3,, xk
• If there exist a relationship, how could we use
  this relationship to forecast future.

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Regression
Regression
Dependent variable
Independent variable (x)

• Regression is the attempt to explain the
  variation in a dependent variable using the
  variation in independent variables.
• Regression is thus an explanation of causation.

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Simple Linear Regression
Regression
y b0 b1X ?
?
Dependent variable (y)
B1 slope ?y/ ?x
b0 (y intercept)
Independent variable (x)

• The output of a regression is a function that
  predicts the dependent variable based upon values
  of the independent variables.
• Simple regression fits a straight line to the
  data.

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Simple Linear Regression
Regression
Observation y

Prediction y
Dependent variable
Zero
Independent variable (x)
The function will make a prediction for each
observed data point. The observation is denoted
by y and the prediction is denoted by y.

For each observation, the variation can be
described as y y
e Actual Explained Error

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Simple Linear Regression

• Simple Linear Regression Model
• y ?0 ?1x ?
• Simple Linear Regression Equation
• E(y) ?0 ?1x
• Estimated Simple Linear Regression Equation
• y b0 b1x

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Simple Linear Regression

• The simplest relationship between two variables
  is a linear one
• y ?0 ?1x
• x independent or explanatory variable (cause)
• y dependent or response variable (effect)
• ?0 intercept (value of y when x 0)
• ?1 slope (change in y when x increases one unit)

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Interpret the slope

• Y0.32.6x

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Regression
Regression
Dependent variable
Independent variable (x)

• A least squares regression, or OLS, selects the
  line with the lowest total sum of squared
  prediction errors.
• This value is called the Sum of Squares of Error,
  or SSE.

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Calculating SSR
Regression
Population mean y
Dependent variable
Independent variable (x)
The Sum of Squares Regression (SSR) is the sum of
the squared differences between the prediction
for each observation and the population mean.
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Regression Formulas
Regression
The Total Sum of Squares (SST) is equal to SSR
SSE. Mathematically, SSR ? ( y y )
(measure of explained variation) SSE ? ( y
y ) (measure of unexplained variation) SST
SSR SSE ? ( y y ) (measure of total
variation in y)

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The Coefficient of Determination
Regression
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Testing for Significance

• To test for a significant regression
  relationship, we must conduct a hypothesis test
  to determine whether the value of b1 is zero.
• t Test is commonly used.

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Testing for Significance t Test

• Hypotheses
• H0 ?1 0
• Ha ?1 0
• Test Statistic
• Rejection Rule Reject H0 if t lt -t????or t gt
  t???? where t??? is based on a t distribution
  with n - 2 degrees of freedom.

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Multiple Linear Regression
Multiple Regression

• More than one independent variable can be used to
  explain variance in the dependent variable.
  
• A multiple regression takes the form
• y A ß X ß X ß k Xk e
• where k is the number of variables, or parameters.

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Multiple Regression
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Regression

• A unit rise in x produces 0.4 of a unit rise in
  y, with z held constant.
• Interpretation of the t-statistics remains the
  same, i.e. 0.4-0/0.41 (critical value is 2.02),
  so we fail to reject the null and x is not
  significant.
• The R-squared statistic indicates 30 of the
  variance of y is explained.

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Adjusted R-squared Statistic

• This statistic is used in a multiple regression
  analysis, because it does not automatically rise
  when an extra explanatory variable is added.
• Its value depends on the number of explanatory
  variables.
• It is usually written as (R-bar squared)

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Adjusted R-squared

• It has the following formula (n-number of
  observations, k-number of parameters)

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F-test of explanatory power

• This is the F-test for the goodness of fit of a
  regression and in effect tests for the joint
  significance of the explanatory variables.
• It is based on the R-squared statistic.
• It is routinely produced by most computer
  software packages
• It follows the F-distribution.

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F-test formula

• The formula for the F-test of the goodness of fit
  is

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F-statistic

• When testing for the significance of the goodness
  of fit, our null hypothesis is that the
  explanatory variables jointly equal 0.
• If our F-statistic is below the critical value we
  fail to reject the null and therefore we say the
  goodness of fit is not significant.

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