a. Suppose that we have data on retail sales (Y_t) by quarters (3 month periods)

Question

a. Suppose that we have data on retail sales (Y_t) by quarters (3 month periods) which are increasing over time. We attempt to explain the increase in retail sales by a regression equation like this: Y_t = beta_0 + beta_1 T + elementof_t where T = 1, 2, 3, 4,...represents time and elementof_t represents errors. How would we modify this equation to seasonally adjust the data to account for things like higher retail sales before Christmas? Write down the equation with any additional variables that would be needed to seasonally adjust the data. b. The equation below shows the results of a regression explaining the satisfaction score of 21 restaurants by the price (X_1) and a dummy variable which is 1 if the restaurant is Italian: [ASW text 15.7.37] The standard errors of the coefficients are given below in parentheses. Y_t = 67.40485 + 0.573429 X_i [0.154574] + 3.03821 D_i [1.15523] R^2 = 0.526233 leftarrow Standard errors of coefficients N = 21 Test whether being an Italian restaurant is significant in increasing the satisfaction score of the restaurant. That is, test H_0: beta_2 lessthanorequalto 0, vs H_A: beta_2 > 0 at alpha = 05. [1-tailed test] c. For the same equation as in 10b, we have an ANOVA table with some values missing: Test whether ALL the independent variables explain Y. That is, test: H_0: beta_1 = beta_2 = 0, vs. H_A: Not all beta_1 = 0, at alpha = 05

Explanation / Answer

b) TS = coeff/se = 3.03821/1.15523 = 2.6299

t-critical for df = n-k-1= 18 is 1.734

since TS> t-critical

we reject the null

c)

df_reg = k = number of independent variable = 2

df_residual = n-k-1 = 18

SS_reg +SS_residual = SS_total

MS = SS/df

F = SS_reg/SS_res

F-critical for df1 = 2 ,df2 = 18 is 3.55

since TS = 9.99 > 3.55

we reject the null and conclude that the model is significant

ANOVA df SS MS F Regression 2 139.5771 69.78855 9.99668871 Residual 18 125.661 6.981166667 Total 20 265.2381

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