- D
The result of estimate function is showed as above, and has value of -3.054252. This means that one unit change in RPRICE 3 will associate in -3.054252% changes in SALT 1. And the 95% confident interval for ΔRPRICE3 is equalled to [-3.054252 -1.96*0.529132, -3.054252 +1.96*0.529132] or approximately [-4.0914, -2.0172], this also means one unit increase by RPRICE 3 will change the SAL1 by between -4.0914 and -2.0172 point, with 95% confident level.
- Similar to part (c), we assume two hypothesises:
Null hypothesis: the slope of the relationship of RPRICE3 and log (SAL1) is zero
Alternative hypothesis: the slope of the relationship of RPRICE3 and log (SAL1) is not zero
According to the table above, the p-value of RPRICE3 is equalled to 0, which is less than 0.01 or 1% significant level. Thus we can reject the Null hypothesis, and this indicates that the relationship between RPRICE3 and log (SAL1) is statistically significant or not zero. As the result from part (d), there is a negative relationship between RPRICE3 and log (SAL1). This also consistent with the economic theory that increase in price of a particular product will encourage consumers to switch to alternative product and result in decrease in unit sales of this particular product.
PART 2
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Creating three new variables: PR1, PR2, and PR3 by using ‘Genr’ function in eview, in term of 100 times the observations on APR1, APR2 and APR3. And estimate the regression model: , we will get the regression table below.
F
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The signs of b2, b3, and b4 are expected following the economic theory, as a result of that there is a negative relationship of no.1 canned tuna and its sales and positive relationships between no.1 canned tuna’s unit sales and its substitute goods price (no.2 and no.3 canned tuna). As increase in price of its substitute goods will encourage consumer to switch their consumption into this particular good.
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For b2, assume H0: =0, H1: <0. With α=0.05 using the p-value approach.
One-side p-value of PR1 = 0.000/2 < 0.05, so reject the null hypothesis. b2 is significantly different away from zero.
For b3, assume H0: =0, H1: >0. With α=0.05 using the p-value approach.
One-side p-value of PR2 = 0.1904/2 > 0.05, so fail to reject the null hypothesis. b3 is not significantly different away from zero.
For b4, assume H0: =0, H1: >0. With α=0.05 using the p-value approach.
One-side p-value of PR3 = 0.0843/2 < 0.05, so reject the null hypothesis. b4 is significantly different away from zero.
- i. By putting C(2)=300 to do the Wald Test, the probability of t-statistic is equalled to 0.0000. This is less than 0.05 or 5% significant level. There we can reject the null hypothesis, A 1-cent increase in the price of brand one can reduces its sales by 300 cans.
ii. By putting C(3)=300 to do the Wald Test, the probability of t-statistic is equalled to 0.0048. This is also less than 0.05 or 5% significant level. We can reject the null hypothesis, A 1-cent increase in the price of brand two increases the sales of brand one by 300 cans. Because the p-value of PR2 is 19% which is not even significant at 10% level, therefore the coefficient of PR2 does not represent the true impact of price change of no.2 canned tuna on unit sales of no.1 canned tuna. However, as the table below shows that PR2, PR3 and PR3 are jointly significant.
iii. By putting C(4)=300 to do the Wald Test, the probability of t-statistic is equalled to 0.1564 > 0.05 or 5% significant level. We fail to reject the null hypothesis; A 1-cent does not increase in the price of brand two increases the sales of brand one by 300 cans.
iv.
The effect of a price increase in brand two on sales of brand one is the same as the effect of a price increase in brand three on sales of brand one. But the outcome of this test is contradicted to my finding from part (iii). the result of part (iii) shows 1 cent increase in the price of brand three does not increase the sales of brand one by 300 cans.
v.
As the table shows, the p-value is 0 < 0.05 or 5% significant level, we should reject the null hypothesis. Therefore, if prices of all 3 brands will goes up by 1 cent, there is no change in sales.
Part 3
By using the same model as part 2 (a), the 95% interval estimate for expected sales when PR1=90, PR2=75 AND PR3=75 is between:
SAL1 = [3246, 42681] + 90*[-631,-311] +75*[-48,234] +75*[-23,353]
= [3246, 42681] + [-56790,-27990] + [-3600, 17550] + [-1725, 26475]
= [-58869, 58716]
These results suggest that one unit increase in PR1, PR2 andPR3 will change SAL1 by between -58869 and 58716. This range does not making sense, because the range is too big and similar. The results do not represent any trend of how unit sales of brand one will be impacted by increasing price of the three brands
By using the new model:,the 95% interval estimate for expected log of sales when PR1=90, PR2=75 and PR3=75 is between:
Log (SAL1) = [8.344072, 12.52782] + 90*[-0.078989,-0.045364] +75*[-0.000618, 0.028966] +75*[0.001682, 0.041262]
= [8.344072, 12.52782] + [-7.10901, -4.08276] + [-0.04635, 2.17245] + [0.12615, 3.09465]
= [1.314862, 13.71216]
The results indicate that one unit increase in PR1, PR2 andPR3 together will change SAL1 by between 1.315% and 13.712%. This is making more sense while comparing this result with part (a). We can conclude that the increase same amount across the three brands will increase the unit sale of brand by at least 1.314%, compare to part (a) that its results can be positive and negative with a similar range.
- The ln(SAL1) as the dependent variable is a log-linear model, which shows one unit change in PR1, PR2 andPR3 are associated with a percentage change in unit sale of brand one(SAL1 ). This is more preferable in showing the price elasticity of a product and effectively showing the trend of unit sales change.
One the other side, the model with SAL1 as the dependent variable is a linear-linear model, which indicates the relationship between one unit changes of PR1, PR2 andPR3 together will result in unit change in SAL1.
Summary:
According to the results above, we can conclude that there is a positive relationship between the sales on brand no.1 canned tuna and the price of the other two brands canned tuna, also a reverse relationship for the price and unit sales of no.1 canned tuna. If Alpha Supermarket stock all three brands canned tuna with expect sales when no.1 =90, no.2 =75 and no.3= 75, one dollar increase in price for all three brands will be 95% sure that the unit sales for no.1 canned tuna will be increased by 1.3% to 13.7%.
