7. Using per capita personal consumption expenditure (PPCE) and per capita disposable income (PPDI)in the United States for the period 1970-1999 to estimate a Koyck model by regressing PPCE on PPDI and lagged PPCE gave the following results: \[ \begin{array}{l} \widehat{P P C E}_{t}=-1242.169+0.6033 P P D I_{t}+0.4106 P P C E_{t-1} . \\ s e=(402.5784) \quad(0.1502) \quad(0.1546) \\ t=(-3.0855) \\ R^{2}=0.9926 \quad D=1.0056 \quad \text { Durbin } h=5.119 \end{array} \] a) Comment on the fit of the model (4 marks) If this model resulted from a Koyck-type transformation, then \( \lambda=0.4106 \). Find the b) Median lag c) Mean lag d) Interpret each of the lags in (a) and (b) above (6 marks)
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The fit of the model is exceptional, highlighted by an \( R^{2} \) value of 0.9926. This indicates that approximately 99.26% of the variation in per capita personal consumption expenditure (PPCE) can be explained by per capita disposable income (PPDI) and lagged PPCE, suggesting a strong correlation. The Durbin-Watson statistic (D=1.0056) is close to 2, indicating little to no autocorrelation in the residuals, strengthening the model's reliability. The median lag can be calculated using the formula \( \frac{1}{1 - \lambda} \), where \( \lambda = 0.4106 \). Plugging in the value, we get \( \text{Median Lag} \approx 1.701 \) periods. For the mean lag, the formula is \( \frac{\lambda}{(1 - \lambda)^2} \). With the same \( \lambda \), the mean lag approximates 2.872 periods. In practical terms, this means that changes in PPDI will have an immediate and significant impact on PPCE, with adjustments continuing to be felt for about 2 to 3 periods.
