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					  Welcome to Econ 420
Applied Regression Analysis
         Study Guide
         Week Eight
                         Answer Key
                   Assignment 6 (20 points)

1.       Suppose the theory suggests that advertising for sun
         blocks is more effective in summer than any other time
         of the year
     –     Formulate the model
           •   Sales = B0 + B1 (Season) + B2 (Advertising) + error
     –     What type of a data set will you use: time series or cross
           sectional?
           •   Time series
     –     Set up a hypothesis to test the theory
     –     The hypothesis of one sided t-test of significance on the
           coefficient of Season is like:
           •   H0: B1 ≤ 0
           •   HA: B1 > 0
           •   If we reject H0, then we have found significant evidence that
               advertising for sun blocks is more effective in summer than any
               other time of the year.
2. Suppose we estimate a regression equation that sets
  the crime rate as a function of a state’s per capita
  income and the number of police officers in each state
  per 10,000 population. The estimated coefficient of per
  capita income happens to be positive. We suspect that
  the estimated coefficient of per capita income is biased
  positively because we have an omitted variable. Which
  of the following omitted variables is more likely to have
  caused the bias in our estimated coefficient of income
  and why?
   –   Number of college educated individuals per 1000 population
   –   Percentage of population living in poverty
   –   State’s unemployment rate
   –   Percentage of population who lives in urban areas.
                                          Answer
•       Direction of bias in B percapita income = Bomitted * r included, omitted
•       Since B percapita income >0
•       Then there are two possibilities
•       (+) = (-) * (-), or
•       (+) = (+) * (+)
    1.       Number of college educated individuals per 1000 population
             can’t be it because it is positively correlated with percapita
             income but it is expected to affect the crime rate negatively
    –        Percentage of population living in poverty cant’ be it because
             it is negatively correlated with percapita income but it is
             expected to affect the crime rate positively.
    –        State’s unemployment rate can’t be it because it is negatively
             correlated with percapita income but it is expected to affect the
             crime rate positively.
    –        Percentage of population who lives in urban areas can be it
             because it is positively correlated with percapita income and it
             is expected to affect the crime rate positively.
        Interaction Variables
• Suppose that the theory suggests that
  advertising is more effective for new
  products than for old products.
• Sales = Bo + B1 (adv) + B2 (adv * age) +
  error
• Where (adv * age) is an interaction
  variable
• Partial derivate of sales with respect to
  advertising is
  – d sales /d adv = B1 + B2 * age
  – Which mean that the advertising affects sales
    depends on the age
  – To find evidence for theory we should find that
    B2 is negative
• How do we set up our hypotheses to test
  the theory?
    Designing your own F test
•   Your model:
•   Y = Bo + B1 X1 + B2 X2 + B3 X3 + B4 X4 +
    e
•   Ho: B2 = B4 = 0
•   HA: at least one is not zero
    Designing your own F test
1. Estimate the model with X1 through X4 in
   it (This is your unrestricted model.) find
   RSS
2. Estimate the model with only X1 and X3
   in it (This is you restricted model) find
   RSS
• Which RSS you expect to be higher?
   Why?
       Designing your own F test
3. Find the critical F
   –   Degrees of freedom for numerator = q = number of
       restrictions
   –   Degrees of freedom for denominator =n-k-1 (k is the
       number of independent variables in the unrestricted
       model)
4. Find Fstat = (RSS restricted – RSS unrestricted )/ q
     divided by RSS unresrticted / n-k-1
5. If Fstat > F critical  reject Ho
              Chow Test
•  Suppose you want to study the
   relationship between hours of study and
   grades and you have two samples
1. 100 MC students
2. 50 Washington State College
• You want to see if it is a good idea to
   combine the two samples
              Chow Test
• Estimate
• (1) GPA = Bo + B1 Study + B2 SAT + B3
  Sleep +e    (for MC, sample n1 = 100)
• (2) GPA = Bo + B1 Study + B2 SAT + B3
  Sleep +e    (for WS, sample n2 = 50)
• (3) GPA = Bo + B1 Study + B2 SAT + B3
  Sleep +e    (for MC & WS, sample n1 + n2
  = 150)
                   Hypotheses
• HO : There is no significant difference between
  Bs from equation 1 and Bs from equation 2
• HA : Ho is not true
• Same F test as before
  –   RSS restricted is the RSS from equation 3
  –   q = number of restrictions = 4 (k +1 in equation 1 or 2)
  –   RSS unrestricted = RSS1 +RSS2
  –   n- k-1 = n1 + n2 – 2 (k+1)= 150-8
         Non-linear models
• The theory suggest that as Q (quantity of
  out put) increases, TR (producer’s total
  revenue) first goes up and then goes
  down.
                          TR curve

     20



     16

                                                  TR
     12
TR




      8



      4



      0
          0   1   2   3        4
                             fig      5   6   7
                           Quantity
• Would this equation capture the theoretical
  shape of TR curve?
• TR = B0 + B1Q + error
• d TR/d Q = B1
• No, B1 can be either positive or negative
  but not both.
      How about this model?
• TR = B0 + B1Q2 + error
• d TR /d Q = 2B1Q
• No, Q is either 0 or positive. So,
  depending on B1, the slope is either
  positive or negative but not both
         How about this one?
• TR = B0 + B1Q + B2 Q2 + e

• dTR/d Q = B1 + 2 B2 Q
• When Q is zero slope is B1
  – So we expect B1 to be positive
• We expect B2 to be negative
  – At high levels of Q, the negative component of the
    slope (2B2Q) will greater that the positive component
    of the slope (B1)
• How do you set up the null and alternative
  hypotheses?
         Double log Models
• Suppose your goal is to estimate the price
  elasticity of demand (E)
• E =% change in Qd divided by % change in
  Price
• ln Qd = B0 + B1 ln P + e
• d ln Qd / d ln P = B1= E
          Semi Log Models
• Suppose you want to estimate the
  percentage growth in a plant as a result of
  1 more teaspoon of fertilizer
• ln Size= B0 + B1Fertilizer
• dlnSize/ d Fertilizer = B1
              Assignment 7
               (40 points)
 Due: Before 10 PM on Friday October 19
1. #4, page 112
2. #5, page 112
3. Use the data set dvd4 and EViews to test
   the hypothesis that at high levels of
   income people are less sensitive to the
   price of dvd than at low levels of income.
   Use 5 percent level of significance.
#13, page 113

				
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posted:4/8/2012
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