1. **Problem Statement:** We have quarterly sales data (in millions) for years 2005 to 2008. We need to: (i) Determine the trend using three-point moving averages. (ii) Estimate sales for each quarter of 2004. (iii) Calculate percentage variation of each quarter's actual sales for 2004. Also, for an AR(2) process $X_t = \frac{1}{3}X_{t-1} + \frac{2}{9}X_{t-2} + e_t$, we need to show stationarity and find autocorrelations $\rho(0), \rho(1), \rho(2)$. --- 2. **Three-Point Moving Average (3MA) Method:** The 3MA smooths data by averaging each point with its neighbors: $$\text{3MA}_t = \frac{X_{t-1} + X_t + X_{t+1}}{3}$$ This helps identify the trend by reducing seasonal fluctuations. --- 3. **Calculate 3MA for each quarter from 2005 Q2 to 2007 Q3:** We treat the data as a sequence of quarters from 2005 Q1 to 2008 Q4 (16 points): $$[33,36,35,38,42,40,42,47,54,53,54,62,70,67,70,77]$$ Calculate 3MA for quarters 2 to 15 (since 3MA requires neighbors): - Q2 (2005 Q2): $\frac{33+36+35}{3} = 34.67$ - Q3 (2005 Q3): $\frac{36+35+38}{3} = 36.33$ - Q4 (2005 Q4): $\frac{35+38+42}{3} = 38.33$ - Q5 (2006 Q1): $\frac{38+42+40}{3} = 40.00$ - Q6 (2006 Q2): $\frac{42+40+42}{3} = 41.33$ - Q7 (2006 Q3): $\frac{40+42+47}{3} = 43.00$ - Q8 (2006 Q4): $\frac{42+47+54}{3} = 47.67$ - Q9 (2007 Q1): $\frac{47+54+53}{3} = 51.33$ - Q10 (2007 Q2): $\frac{54+53+54}{3} = 53.67$ - Q11 (2007 Q3): $\frac{53+54+62}{3} = 56.33$ - Q12 (2007 Q4): $\frac{54+62+70}{3} = 62.00$ - Q13 (2008 Q1): $\frac{62+70+67}{3} = 66.33$ - Q14 (2008 Q2): $\frac{70+67+70}{3} = 69.00$ - Q15 (2008 Q3): $\frac{67+70+77}{3} = 71.33$ --- 4. **Estimate sales for 2004 quarters:** Assuming trend continues backward, use 3MA to estimate 2004 Q1 to Q4. We extrapolate using first few 3MA values: - 2004 Q4 estimate = 3MA at 2005 Q1 (Q1 is 33, no 3MA, so average first 3 points): $\frac{33+36+35}{3} = 34.67$ - 2004 Q3 estimate = average of 2004 Q2, Q4, and 2005 Q1 (approximate as 34.67) Since data before 2005 is unavailable, we estimate 2004 quarters as approximately 30, 31, 32, 33 million (rough estimate based on trend growth). --- 5. **Percentage variation for 2004 quarters:** Percentage variation = $\frac{\text{Actual} - \text{Estimated}}{\text{Estimated}} \times 100$% Since actual 2004 data is not given, this cannot be computed exactly. --- 6. **AR(2) Process Stationarity:** Given: $$X_t = \frac{1}{3}X_{t-1} + \frac{2}{9}X_{t-2} + e_t$$ The characteristic equation is: $$1 - \frac{1}{3}z - \frac{2}{9}z^2 = 0$$ Multiply both sides by 9: $$9 - 3z - 2z^2 = 0$$ Rewrite: $$2z^2 + 3z - 9 = 0$$ Solve for $z$: $$z = \frac{-3 \pm \sqrt{9 + 72}}{4} = \frac{-3 \pm 9}{4}$$ Roots: $$z_1 = \frac{6}{4} = 1.5, \quad z_2 = \frac{-12}{4} = -3$$ Stationarity requires roots of characteristic equation to lie outside unit circle in $z$-plane (i.e., $|z| > 1$). Here, $|1.5| > 1$ and $|3| > 1$, so process is stationary. --- 7. **Autocorrelation Function (ACF):** The Yule-Walker equations for AR(2) are: $$\rho(1) = \phi_1 + \phi_2 \rho(1)$$ $$\rho(2) = \phi_1 \rho(1) + \phi_2$$ Given $\phi_1 = \frac{1}{3}$, $\phi_2 = \frac{2}{9}$. From the first equation: $$\rho(1) = \frac{1}{3} + \frac{2}{9} \rho(1) \Rightarrow \rho(1) - \frac{2}{9} \rho(1) = \frac{1}{3}$$ $$\rho(1) \left(1 - \frac{2}{9}\right) = \frac{1}{3} \Rightarrow \rho(1) \times \frac{7}{9} = \frac{1}{3}$$ $$\rho(1) = \frac{1}{3} \times \frac{9}{7} = \frac{3}{7} \approx 0.4286$$ Then: $$\rho(2) = \frac{1}{3} \times \frac{3}{7} + \frac{2}{9} = \frac{1}{7} + \frac{2}{9} = \frac{9}{63} + \frac{14}{63} = \frac{23}{63} \approx 0.3651$$ Also, by definition: $$\rho(0) = 1$$ --- **Final answers:** - 3MA trend values calculated for quarters 2005 Q2 to 2008 Q3. - Estimated 2004 sales roughly extrapolated. - Stationarity confirmed for AR(2) process. - Autocorrelations: $\rho(0) = 1$, $\rho(1) = \frac{3}{7}$, $\rho(2) = \frac{23}{63}$.