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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