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A ski resort has come up with a model to predict the number of guests (given in hundreds) checking in throughout the year. The time series model contains both a trend and a seasonal component and is given by the following:

y_t = 0.2 \times \text{Time}_t + 10.5 + 3.0 \times D_{2t} + 2.1 \times D_{3t} + 2.8 \times D_{4t}

The trend component is reflected in variable $\text{TIME}_{(t)}$ , where $(t) = \text{month}$ .

Seasons are defined as follows:

Season	Months	Dummy
Winter	December, January and February	—
Spring	March, April and May	$D_{2t}$
Summer	June, July and August	$D_{3t}$
Fall	September, October and November	$D_{4t}$

The model starts in April 2019, i.e., $y_{(T+1)}$ refers to May 2019. How many more guests are expected in April 2020 than in July of the same year?

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TTanishq

A

10

B

30

C

9

D

15

Explanation:

Explanation

The model is given as:

y_t = 0.2 \times \text{Time}_t + 10.5 + 3.0 \times D_{2t} + 2.1 \times D_{3t} + 2.8 \times D_{4t}

Since we have three dummies and an intercept, quarterly seasonality is reflected by the intercept (10.5) plus the three seasonal dummy variables ( $D_2$ , $D_3$ , and $D_4$ ).

If $y_{(T+1)} = \text{May 2019}$ , then:

April 2020 = $y_{(T+12)}$
July 2020 = $y_{(T+15)}$

Note that April falls under $D_{(2t)}$ (Spring season), while July falls under $D_{(3t)}$ (Summer season).

Calculation for April 2020 ( $y_{(T+12)}$ ):

Time component: $0.2 \times 12 = 2.4$
Intercept: $10.5$
Spring dummy: $3.0 \times 1 = 3.0$
Total: $2.4 + 10.5 + 3.0 = 15.9$

Calculation for July 2020 ( $y_{(T+15)}$ ):

Time component: $0.2 \times 15 = 3.0$
Intercept: $10.5$
Summer dummy: $2.1 \times 1 = 2.1$
Total: $3.0 + 10.5 + 2.1 = 15.6$

Difference:

y_{(T+12)} - y_{(T+15)} = 15.9 - 15.6 = 0.3

Since the number of guests is given in hundreds, the actual difference is: $`$ 0.3` \times 100 = 30 \text{ guests}$$

Thus, the model predicts 1,590 guests in April and 1,560 guests in July, representing a difference of 30 guests.

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