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The seasonal dummy model is generated on the quarterly growth rates of mortgages. The model is given by:

Y_t = \beta_0 + \sum_{j=1}^{s-1} \gamma_j D_{jt} + e_t

The estimated parameters are $\hat{\gamma}_1 = 6.25$ , $\hat{\gamma}_2 = 50.52$ , $\hat{\gamma}_3 = 10.25$ and $\hat{\beta}_0 = -10.42$ using the data up to the end of 2019. What is the difference between the forecasted value of the growth rate of the mortgages in the second and third quarters of 2020?

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NNikitesh

Last updated: February 2, 2026 at 10:22

A

24.56

B

32.45

C

40.27

D

30.32

Explanation:

Explanation

This is a seasonal dummy model with quarterly data. The model includes:

$\beta_0$ : Intercept term
$\gamma_j$ : Seasonal coefficients for quarters (with s-1 = 3 dummies, meaning quarter 4 is the base/reference quarter)
$D_{jt}$ : Dummy variables for quarters 1, 2, and 3

Step 1: Forecast for Q2 2020 For Q2, the dummy variables are:

$D_{1t} = 0$ (not Q1)
$D_{2t} = 1$ (Q2)
$D_{3t} = 0$ (not Q3)

Forecast: $E(\hat{Y}_{Q_2}) = \beta_0 + \gamma_1 \times 0 + \gamma_2 \times 1 + \gamma_3 \times 0$ $= -10.42 + 0 \times 6.25 + 1 \times 50.52 + 0 \times 10.25$ $= -10.42 + 50.52 = 40.10$

Step 2: Forecast for Q3 2020 For Q3, the dummy variables are:

$D_{1t} = 0$ (not Q1)
$D_{2t} = 0$ (not Q2)
$D_{3t} = 1$ (Q3)

Forecast: $E(\hat{Y}_{Q_3}) = \beta_0 + \gamma_1 \times 0 + \gamma_2 \times 0 + \gamma_3 \times 1$ $= -10.42 + 0 \times 6.25 + 0 \times 50.52 + 1 \times 10.25$ $= -10.42 + 10.25 = -0.17$

Step 3: Calculate the difference $Q_2 - Q_3 = 40.10 - (-0.17) = 40.10 + 0.17 = 40.27$

Therefore, the difference between the forecasted growth rates for Q2 and Q3 2020 is 40.27, which corresponds to option C.

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