To develop a multiple regression model that accounts for both trend and seasonal effects in your time series data, follow these steps:

1. Define the variables:

   • Response variable ($\hat{y}$): This will be your dependent variable, which represents the data you provided.
   • Trend variable ($t$): This is a time variable where $t$ = 1 for Quarter 1 of Year 1, $t$ = 2 for Quarter 2 of Year 1, and so forth until $t$ = 12 for Quarter 4 of Year 3.
   • Dummy variables for seasons: Create three dummy variables for the quarters:
     • $\text{Qtr1}$: 1 if Quarter 1, 0 otherwise.
     • $\text{Qtr2}$: 1 if Quarter 2, 0 otherwise.
     • $\text{Qtr3}$: 1 if Quarter 3, 0 otherwise.
     • $\text{Qtr4}$: 1 if Quarter 4, 0 otherwise.

2. Construct the model: The general form of your regression equation will be:

   $\hat{y} = \beta_0 + \beta_1 \cdot \text{Qtr1} + \beta_2 \cdot \text{Qtr2} + \beta_3 \cdot \text{Qtr3} + \beta_4 \cdot t$

3. Data Setup: Using the provided data, structure it for analysis:

   • For Year 1:
     • Q1: 2 (t = 1, Qtr1 = 1, Qtr2 = 0, Qtr3 = 0, Qtr4 = 0)
     • Q2: 4 (t = 2, Qtr1 = 0, Qtr2 = 1, Qtr3 = 0, Qtr4 = 0)
     • Q3: 1 (t = 3, Qtr1 = 0, Qtr2 = 0, Qtr3 = 1, Qtr4 = 0)
     • Q4: 7 (t = 4, Qtr1 = 0, Qtr2 = 0, Qtr3 = 0, Qtr4 = 1)
   • For Year 2:
     • Q1: 4 (t = 5, Qtr1 = 1, Qtr2 = 0, Qtr3 = 0, Qtr4 = 0)
     • Q2: 5 (t = 6, Qtr1 = 0, Qtr2 = 1, Qtr3 = 0, Qtr4 = 0)
     • Q3: 3 (t = 7, Qtr1 = 0, Qtr2 = 0, Qtr3 = 1, Qtr4 = 0)
     • Q4: 9 (t = 8, Qtr1 = 0, Qtr2 = 0, Qtr3 = 0, Qtr4 = 1)
   • For Year 3:
     • Q1: 5 (t = 9, Qtr1 = 1, Qtr2 = 0, Qtr3 = 0, Qtr4 = 0)
     • Q2: 8 (t = 10, Qtr1 = 0, Qtr2 = 1, Qtr3 = 0, Qtr4 = 0)
     • Q3: 4 (t = 11, Qtr1 = 0, Qtr2 = 0, Qtr3 = 1, Qtr4 = 0)
     • Q4: 10 (t = 12, Qtr1 = 0, Qtr2 = 0, Qtr3 = 0, Qtr4 = 1)

4. Fit the regression model: You can use statistical software (like R, Python's statsmodels, or Excel) to fit the regression model and obtain coefficients $\beta_0, \beta_1, \beta_2, \beta_3, \beta_4$.

5. Example output format: After performing the regression analysis, you might get an output like:

   $\hat{y} = 2.000 + 1.500 \cdot \text{Qtr1} + 0.800 \cdot \text{Qtr2} + -0.300 \cdot \text{Qtr3} + 0.200 \cdot t$

   (Note: These values are placeholders; you would need to calculate the actual coefficients from your data.)

Conclusion

Make sure to check for the significance of your coefficients and the overall model fit (using R-squared and residual analysis).

If you need further assistance with the regression analysis or interpreting results, let me know!

Follow-Up Questions

1. What software or tool are you using for the regression analysis?
2. Would you like guidance on how to interpret the regression coefficients?
3. Do you need help with checking the assumptions of the regression model?
4. Are you interested in forecasting future values based on this model?
5. Would you like to explore alternative modeling techniques, such as ARIMA or exponential smoothing?

Tip

Always check for multicollinearity among your independent variables, as this can distort the regression results.