math and econ..
math and econ. 1. (a) Clearly and precisely define the concepts œhomoscedasticity and œheteroscedasticity. Be sure to highlight any differences between the two concepts. (Feel free to use diagrams if they can be used to clarify your answer). (15 marks) (b) When testing multiple linear restrictions in multiple regression analysis framework, there are two formula’s that are used to calculate the F test statistic used to test the model. Firstly we are given (1) ( 1) ( ) ( , 1) “ – – “ “ = n k SSR q SSR SSR F ur r ur q n k 1 and secondly (2) ( 1) (1 ) ( ) 2 2 2 ( , 1) “ – – – “ “ = n k R q R R F ur ur r q n k Note: F(q,n-k-1) simply means that we must check the test statistics against the F distribution with q numerator degrees of freedom and n-k-1 denominator degrees of freedom. r SSR and SSRur are the sum of squared residuals from the restricted and unrestricted models respectively. 2 Rr and 2 Rur are the 2 R from the restricted and unrestricted models respectively. Can you prove that equation (1) is exactly the same as equation (2)? (15 marks) 2) The file contains 56 observations on variables related to the sales of cocaine powder in northeastern California over the period 1984-1991. The data are a subset of those used in the study Caulkins, J.P. and R. Padman (1993), œQuantity Discounts and Quality Premia for Illicit Drugs, œJournal of the American Statistical Association, 88, 748-757. The variables are PRICE = Price per gram in dollars for a cocaine sale QUANT = number of grams of cocaine in a given sale QUAL = Quality of the cocaine expressed as percentage purity Trend = A time variable with 1984 = 1 up to 1991 = 8 Consider the regression model: ???????????????????? = ????1 + ????2???????????????????? + ????3???????????????? + ????4???????????????????? + ???? (a) What signs would you expect on the coefficients ????2, ????3 and ????4? (Provide a justification) (5 marks) (b) Use EViews to estimate the equation. Report the results and interpret the coefficient estimates. Have the signs turned out as you expected? (5 marks) (c) What proportion of the variation in the cocaine price is explained jointly by variation in quantity, quality and time? (5 marks) 2(d) It is claimed that in this business the greater the number of sales the higher the risk of getting caught. This implies that sellers are willing to accept a lower price if they can make sales in large quantities. Set up ????0 and ????1 that would be appropriate to carry out the test of hypothesis and conduct the test. (5 marks) (e) Test the hypothesis that the quality of cocaine has no influence on the price against the alternative that a premium is paid for a better quality of cocaine. (8 marks) (f) What is the average annual change in the cocaine price? Can you suggest a plausible explanation for why the prices have been changing in that direction? (7 marks) (3) In Table 1 below, the regression output of three models is presented. Table 1 Dependent variables: lprice Independent variables Model 1 Model 2 Model 3 Lasses 1.0431 (0.514) 0.97163 (0.06773) 1.0220 (0.1528) Lsqrft 0.10324 (0.05384) 0.09532 (0.01789) Bdrms -0.033839 (0.01551) -0.028372 (0.01610) -0.09133 (0.01044) Llotsize 0.0074379 (0.03856) 0.06945 (0.02567) bdrms2 0.00423 (0.00245) 0.013055 (0.00566) Intercept 0.26374 (0.4630) -0.023814 (0.3593) 0.42294 (0.5900) Observations 80 80 80 R-squared 0.8044 0.7005 0.7657 Sum of squared residuals (SSR=?u2 ) 1.9215 1.8399 1.4282 The dependent variable lprice is the natural logarithm of house prices (GBP£’000). The explanatory variables are as follows: lassess is the natural logarithm of the value of the house as assessed by the city council (£’000), lsqrft is the natural logarithm of the size of the house measured in squared feet, bdrms is the number of bedrooms and llotsize is the natural logarithm of the size of the land section measured in squared feet. Standard errors are in parenthesis. Based on the information in Table 1 answer the following questions: (a) Using the results from Model 2 test the null hypothesis that lsqrft and 2 bdrms have no effect on house prices in Model 3. (10 marks) 3(b) Test the null hypothesis that none of the explanatory variables in Model 1 has any effect on house prices. (10 marks) (c) What is the partial effect of bdrms on house prices in Model 1? Describe the relationship between these variables. Sketch a schedule that illustrates this relationship and provide and explanation for the shape of the schedule. (15 marks)