2. Suppose you are interested in learning if there are seasonal patterns in part

Question

2. Suppose you are interested in learning if there are seasonal patterns in party supply sales in OC, that is, you would like to know if sales in a particular season are significantly different from other seasons. Suppose the following model is estimated: where sales is the quarterly party supply sales in thousands of dollars in OC; D, Da, and Ds are dummies indicating the first, the second, and the fourth quarter respectively. (1) How would you interpret the estimated intercept and slopes? (2) How would you test if sales in the third quarter are significantly different from those in the fourth quarter? Write down the null hypothesis and decide whether you need to do a tor an F test.

Explanation / Answer

Consider the given problem, here we have a model which is given by.

Sales = a + b1*D1 + b2*D2 + b3*D3, where “Di” be dummies, “Di=1 represent the presence of “ith” quarter and “0” represent the absence of that quarter.

So, here we can divide 1 entire year into 4 quarters, so here “D1” represent quarter 1, “D2” represent quarter 2, “D4” represent quarter 4.

So, here “D3” be the base quarter here.

1).

So, if we estimate the given model then, the model then the intercept shows the “mean sales in quarter 3”, and all the coefficient represent the intercept differential for ith quarter

So, “a+b1” represent the mean sales in quarter 1, so “b1” the coefficient represent the sales differential from the base quarter 3.

Similarly, “a+b2” represent the mean sales in quarter 2, so “b2” the coefficient represent the sales differential from the base quarter 3.

Similarly, “a+b4” represent the mean sales in quarter 4, so “b4” the coefficient represent the sales differential from the base quarter 3.

2).

Now, if we want to test that the sales of quarter3 is significantly differ from quarter4, so here we need to test for “b4”.

So, here the “H0:b4=0”, and the alternative hypothesis be “H1:b4 not=0”.

So, given this the corresponding statistic be “t=b4/SE(b4)”.

So, if “mod(t)” > the critical value of “t” at the chosen level of significance, the “H0” will rejected, and we can say that the mean sales in quarter3 is significantly differ than quarter4.

3).

If we test for the seasonality then we need to perform “F” test.

So, “H0:b1=b2=b4=0” and the corresponding alternative hypothesis is “H1: they are differ form 0”.

Now, if the calculated value of “F” is more than the critical value of “F” at the chosen level of significance, the “H0” will be rejected, and we can say that the mean sales of quarters are significantly differ in each other. When “H0” will be accepted => there is no seasonality => the mean sales in all the quarter are more or less same.

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