ECONOMETRICS Part I Now, you are required to do the analysis for the weekly sales of a major brand of

ECONOMETRICS

Part I

Now, you are required to do the analysis for the weekly sales of a major brand of canned tuna, Brand #1, by a supermarket chain store in a large Midwestern U.S city during a mid-2000's calendar year.

According to the Consumer Theory, you know that the quantity of a good will not only depend on its own price but also relative goods' price. Furthermore, you know according to the Consumer Behavior Theory, promotion will enhance and establish consumers' preferences to buy more. Meaning the promotion, say advertisement (ad), will attract the consumers to buy more, increasing sales. From the above information, you would expect that the quantity of Tuna sales (Brand #1) per week, Sale 1, will be affected by its own price, Price 1 , and the other canned tune brand competitors' prices, Price 1 and Price 2. Furthermore, if the store tries to promote Brand #1 it will increase its sells, and the store manager has options to promote either through display ads in the store, or promote in BOTH the store and in newspapers ads, or do neither.

Table 1 Variable Definition

Sale $1 \quad$ number of cans of brand $\# 1$ sold during week

Price $1 \quad$ average price per can of brand $\# 1$ during week

Price $2 \quad$ average price per can of brand $\# 2$ during week

Price $3 \quad$ average price per can of brand $\# 3$ during week

Disp $\quad=1$ if there is a store display for brand #1 during week but no newspaper ad,

$=0$ Otherwise

Dispad $\quad=1$ if there is a store display for brand #1 during week AND a newspaper ad during week,

$=0$ Otherwise

Summer $\quad=1$ during June to August $\quad=0$ Otherwise

Fall $=1$ during September to December $=0$ Otherwise

Winter $=1$ during January to March $=0$ Otherwise

Based on above information, you decide to build up your models as follows, first:

Model A

Sale $1=B_{0}+B_{1}$ Price $1+B_{2}$ Price $2+B_{3}$ Price $3+B_{4}$ disp $+B_{5}$ dispad $+u$

[5 points] Report you results of the estimated regression in the form of an equation. Write your results listing the standard errors, t-value and $P$ -value in parentheses under the numerical estimates of each estimator. Using stars to indicate $10 \%$ significance levels, and listing the $R^{2}$ and $\bar{R}^{2}$.
[5 points ] Discuss and interpret the estimates of $B_{1}, B_{2}$, and $B_{3}$ for $\operatorname{Model} A$.
[5 points] How do you interpret the coefficients on the dummies in this model?
[5 points] Which of the variables are statistically significant? Are the signs of these estimates consistent with economic logic?
[10 points ]Test, at the $\alpha=10 \%$ level of significance, each of the following hypotheses: (Be sure to state: the null and alternative hypotheses, the critical value of the test statistic, degree of freedom, the computed value of the test statistic, the criterion for acceptance or rejection of the null hypothesis, and your conclusion.)

$H_{0}: B_{1}>0 \quad H_{0}: B_{1} \leq 0$
$H_{0}: B_{5}=0 \quad H_{0}: B_{5} \neq 0$
$H_{0}: B_{4}=B_{5}=0 \quad H_{0}: B_{4}$ or $B_{5} \neq 0$

6. [10 points] According to the above question $5 b$ and $5 c$, interpret the results to explain the proper decision of the supermarket.

There is a famous marketing director, Dr. Fish, that suggests that you should use $\ln ($ Sale 1 ) as dependent. According to his experience, he believes the relationship between sales and price is not a constant. You decide to follow his suggestions and create a new variable, $\ln$ _sale_1, and estimate again your new model, Model B

Model B

$\ln (\text { Sale } 1)=B_{0}+B_{1} \text { Price } 1+B_{2} \text { Price } 2+B_{3} \text { Price } 3+B_{4} \text { disp }+B_{5} \text { dispad }+u$

7. [10 points] Report you results of the estimated regression in the form of an equation for Model B. Write your results listing the standard errors, t-value and $\mathrm{P}$ -value in parentheses under the numerical estimates of each estimator. Using stars to indicate $10 \%$ significance levels, and listing the $R^{2}$ and $\bar{R}^{2}$.

8. [5 points] Discuss and interpret the estimates of $B_{1}, B_{2}$, and $B_{3}$ for Model $B$.

There is another famous marketing director, Dr. Doolittle, that suggests that you should include seasonal patterns; Spring, Summer, Fall and Winter in your model. You decide to follow his suggestions and generate the three new variables, say Summer, Fall and Winter, and estimate again your new model, Model C. What is the rationale for the including only three seasons, say: Summer, Fall and Winter? Please explain the signs of these new variables.

Model C

\(\begin{aligned}

& \ln (\text{ Sale }1)={{B}_{0}}+{{B}_{1}}\text{ Price }1+{{B}_{2}}\text{ Price }2+{{B}_{3}}\text{ Price }3+{{B}_{4}}\text{ disp }+{{B}_{5}}\text{ dispad }+{{B}_{6}}\text{ Summer} \\

& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{ }+{{B}_{7}}\text{ Fall }+{{B}_{8}}W\text{ int }er+u \\

\end{aligned}\)

9. [10 points] Test for Model $C$ at the $\alpha=10 \%$ level of significance, There is no structural change at all over the seasons against the alternative that Summer, Fall, and Winter were structurally different, allowing the types of promotion to vary. Be sure to state: the null and alternative hypotheses, the critical value of the test statistic, degree of freedom, the computed value of the test statistic, the criterion for acceptance or rejection of the null hypothesis, and your conclusion.)

10. [5 points]Using Model $C$ predict the amount of sales for Brand $\# 1$ in summer with ad in store AND in the newspaper with Price $2=0.75$ Price $2=0.67$ and Price $3=0.83$

11. [10 points] Summary of your findings from Model A, Model B and Model C. Also, according to your analysis above which model provides the most reasonable results.

12. [5 points] Discuss the internal validity of the Model C.

Part II

13. [5 points] Look over the following equations and decide whether they are linear in parameters or not.