Module 2Demand Theory and Estimation

Activity 4Case 3

Using SPSS, MS Excel or other statistical software package, run the given data and answer the following in MS Word.

Assume that Electronix, Inc., a small startup company that distributes a particular business machine, has the following monthly data on unit sales (Q), price (P), advertising expenditures (AD), and personal selling expenditures (PSE) over the past year.

If a linear relation between unit sales, price, advertising, and personal selling expenditures is hypothesized, the regression equation takes the following form: where is the number of units sold, P is the average price per month, D is advertising expenditures, PSE is personal selling expenditures, and is a random disturbance term - all measured on a monthly basis over the past year.

B. Estimate the regression equation of on the explanatory variables.

C. Check for consistency in the relationship between quantity demand and the three explanatory variables as postulated in demand theory by indicating the change in the quantity demanded of the commodity for each unit change in the explanatory variables.

Show all results in two decimal places.

MonthUnits SalesAdvertising Personal Selling

Price ($) Expenditures ($) Expenditures ($)

January2,5003,80026,80043,000

February2,2503,70023,50039,000

March1,7503,60017,40035,000

April1,5003,50015,30034,000

May1,0003,20010,40026,000

June2,5003,20018,40041,000

July2,7503,20028,20040,000

August1,7503,00017,40033,000

September1,2502,90012,30026,000

October3,0002,70029,80045,000

November2,0002,70020,30032,000

December2,0002,60019,80034,000

Problem 2. Give an example of a stochastic process that is: (a) Both a Markov process and a martingale. (b) A Markov process but not a martingale. (c) A martingale but not a Markov process. (d) Neither a Markov process nor a martingale.2. A Markov chain with state space [1, 2, 3} has transition probability matrix 00 0.3 0.1 10\": 0.3 0.3 0.4 0.4 0.1 0.5 (a) Is this Markov chain irreducible? Is the Markov chain recurrent or transient? Explain your answers. (b) What is the period of state 1? Hence deduce the period of the remaining states. Does this Markov chain have a limiting distribution? (0) Consider a general three-state Markov chain with transition matrix P11 P12 P13 P = P21 P22 P23 031 P32 P33 Give an example of a specic set of probabilities pg'j for which the Markov chain is not irreducible (there is no single right answer to this1 of course l]. 2. (15 pts) Consider a Markov chain { Xn } with state space S = {0, 1, 2} and transition matrix and transition matrix P = O ONIH HNIH O (1) Let the mapping f : S - S satisfy f(0) = 0 and f(2) = 1 and assume that f(1) + f(2). If Yn = f(Xn), then when is { Yn } a Markov chain? Is {Yn } always a Markov chain? In other words, are functions of Markov chains always Markov chains?Given a Markov chain with transition matrix P and stationary distribution I, the time reversal is a Markov chain with transition matrix P defined by Pi = for all i.j. (a) Show that a Markov chain with transition matrix P is reversible if and only if P = P. (b) Show that the time reversal Markov chain has the same stationary distribu- tion as the original chain.Suppose the variance ,2 and the mean / both are unknown. Let the sample mean of a large n Li.d. observations be x and sample variance be $2. What could be the distribution of -#? V52 Select one: O a. Gaussian with mean / and variance g? O b. Gaussian with mean o and variance 1 O c. Gaussian with mean o and variance reduced to o O d. Since vse is now another random variable, the given is now a ratio of two random variables, which is likely not Gaussian O e. Gaussian with mean o and variance g