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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To analyze the Koyck model results provided, we will address each part of the question step by step.

### a) Comment on the fit of the model

The fit of the model can be assessed using the $$ R^2 $$ value and the Durbin-Watson statistic.

1. **$$ R^2 = 0.9926 $$**: This indicates that approximately 99.26% of the variation in per capita personal consumption expenditure (PPCE) can be explained by the model, which includes per capita disposable income (PPDI) and lagged PPCE. This is an exceptionally high $$ R^2 $$, suggesting a very good fit.

2. **Durbin-Watson statistic $$ D = 1.0056 $$**: This statistic tests for autocorrelation in the residuals. A value close to 2 suggests no autocorrelation, while values below 1 or above 3 indicate positive or negative autocorrelation, respectively. A value of 1.0056 suggests that there may be positive autocorrelation present in the residuals, which could indicate that the model might not fully capture the dynamics of the data.

3. **Standard Errors and t-values**: The standard errors for the coefficients are relatively low, indicating that the estimates are precise. The t-value for the intercept is -3.0855, which suggests that it is statistically significant. The coefficients for PPDI and lagged PPCE are also likely significant given their standard errors.

Overall, while the model fits the data well in terms of $$ R^2 $$, the Durbin-Watson statistic indicates potential issues with autocorrelation, which should be addressed in further analysis.

### b) Median lag

The median lag in a Koyck model can be calculated using the formula:

$$
\text{Median Lag} = \frac{\ln(2)}{\lambda}
$$

Given $$ \lambda = 0.4106 $$:

$$
\text{Median Lag} = \frac{\ln(2)}{0.4106}
$$

Let's calculate this value.

### c) Mean lag

The mean lag in a Koyck model can be calculated using the formula:

$$
\text{Mean Lag} = \frac{1}{1 - \lambda}
$$

Using $$ \lambda = 0.4106 $$:

$$
\text{Mean Lag} = \frac{1}{1 - 0.4106}
$$

Let's calculate this value as well.

### d) Interpret each of the lags in (b) and (c) above

1. **Median Lag**: The median lag represents the time it takes for half of the effect of a change in PPDI on PPCE to be realized. A shorter median lag indicates that the impact of changes in disposable income on consumption is felt relatively quickly.

2. **Mean Lag**: The mean lag represents the average time it takes for the full effect of a change in PPDI on PPCE to be realized. It provides a more comprehensive view of the lag structure, accounting for all periods of influence. A longer mean lag suggests that the effects of changes in disposable income on consumption are spread out over a longer period.

Now, let's calculate the median and mean lags.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{1}{\left(1-0.4106\right)}$$
- step1: Remove the parentheses:
  $$\frac{1}{1-0.4106}$$
- step2: Subtract the numbers:
  $$\frac{1}{0.5894}$$
- step3: Convert the expressions:
  $$\frac{1}{\frac{2947}{5000}}$$
- step4: Multiply by the reciprocal:
  $$\frac{5000}{2947}$$
Calculate or simplify the expression $$ \ln(2)/0.4106 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\ln{\left(2\right)}}{0.4106}$$
- step1: Rewrite the expression:
  $$\frac{\ln{\left(2\right)}}{\frac{2053}{5000}}$$
- step2: Multiply by the reciprocal:
  $$\ln{\left(2\right)}\times \frac{5000}{2053}$$
- step3: Multiply the terms:
  $$\frac{\ln{\left(2\right)}\times 5000}{2053}$$
- step4: Multiply the terms:
  $$\frac{5000\ln{\left(2\right)}}{2053}$$
### b) Median Lag

The median lag is calculated as follows:

$$
\text{Median Lag} \approx 1.6881
$$

This means that it takes approximately 1.688 years for half of the effect of a change in per capita disposable income (PPDI) on per capita personal consumption expenditure (PPCE) to be realized.

### c) Mean Lag

The mean lag is calculated as follows:

$$
\text{Mean Lag} \approx 1.6966
$$

This indicates that, on average, it takes about 1.697 years for the full effect of a change in PPDI on PPCE to be realized.

### d) Interpretation of the Lags

1. **Median Lag (1.6881 years)**: This value suggests that when there is a change in disposable income, approximately half of the impact on consumption is felt within about 1.688 years. This relatively short duration indicates that consumers respond fairly quickly to changes in their disposable income.

2. **Mean Lag (1.6966 years)**: The mean lag of approximately 1.697 years indicates that the total effect of a change in disposable income on consumption is spread out over this period. This means that while the initial response is quick, the full adjustment to the new level of income takes a bit longer, reflecting the gradual nature of consumption adjustments.

In summary, both the median and mean lags suggest that consumers in the U.S. adjust their consumption in response to changes in disposable income within a relatively short time frame, which is consistent with economic theories regarding consumer behavior.
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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