Question

Answer 1

I can provide a general approach to answering the given questions:

a. To fit a regression model with a time trend and seasonal dummy variables to the sales data, you would need to first identify the seasonal pattern in the data (e.g., quarterly, monthly, weekly) and create dummy variables to capture this pattern. You would then include these dummy variables along with a time trend variable (e.g., the quarter or month number) in a linear regression model, where the sales data is the dependent variable and the time trend and dummy variables are the independent variables.

b. To determine if the time trend coefficient is statistically significant, you would need to perform a hypothesis test on the coefficient using a t-test or other appropriate statistical test. The null hypothesis would be that the time trend coefficient is equal to zero (i.e., there is no linear trend in the sales data), and the alternative hypothesis would be that the coefficient is not equal to zero (i.e., there is a linear trend in the sales data). If the p-value associated with the test is less than the chosen significance level (e.g., 0.05), then the time trend coefficient is considered statistically significant.

c. To determine if the seasonal dummy variables are statistically significant, you would perform a similar hypothesis test on each dummy variable, where the null hypothesis is that the coefficient associated with the variable is equal to zero (i.e., there is no seasonal effect on sales for that period) and the alternative hypothesis is that the coefficient is not equal to zero (i.e., there is a seasonal effect on sales for that period). If the p-value associated with the test is less than the chosen significance level, then the seasonal dummy variable is considered statistically significant.

d. To calculate sales for Q3 with time 0, you would need to know the functional form of the regression model (e.g., linear, quadratic) and the values of the time trend and dummy variables for Q3 when time is 0. You would then plug these values into the regression equation to obtain the predicted sales value for Q3.

e. To obtain the coefficient on the first quarter dummy variable, you would simply look at the estimated coefficient for that variable in the regression output. This coefficient represents the change in sales associated with the first quarter (relative to the baseline period, which is usually the last quarter or the period with the lowest sales) after controlling for the time trend and other factors included in the model.