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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)

Ask by Stephens Lane. in Kenya
Mar 12,2025

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The model fits the data very well, explaining 99.26% of the variation. However, there is some positive autocorrelation in the residuals. The median lag is approximately 1.688 years, meaning half of the effect of disposable income on consumption is felt within about 1.688 years. The mean lag is about 1.697 years, indicating that the full effect takes roughly 1.697 years to be realized. These lags suggest that consumers adjust their spending in response to income changes relatively quickly, with the full adjustment occurring over a slightly longer period.

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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.

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