Garden City Plants is planning its full-sized tree purchases for the next several years, and is using the model of tree demand developed from past experience. The manager is responsible for developing the tree demand model. Garden City Plants is the leader in the community, and other suppliers follow its plans.
The model the manager uses is the following, estimated on 34 years of past data:
0 = 450 - 5P - 225 + 3Y + 12H R² = 844
(17) (.13) (.16) (.08) (6.22)
(standard errors are in parentheses)
where Qt = full-sized tree demand in year t, Pt = price of trees in year t, St = price of saplings in year t, Yt = per capita income in the
metro area in year t, in thousands of dollars, and Ht = average housing price in the metro area in year t, in thousands of dollars.
a. Evaluate these regression results, including computation of t-statistics, coefficients and R².

To evaluate the regression results for Garden City Plants' tree demand model, we need to analyze the regression coefficients, compute the t-statistics for each coefficient, and evaluate the R-squared value. Here's a step-by-step breakdown:

### Given Regression Model:
\[ Q_t = 450 - 5P_t - 225S_t + 3Y_t + 12H_t \]
\( R^2 = 0.844 \)

Where:
- \( Q_t \) = full-sized tree demand in year \( t \)
- \( P_t \) = price of trees in year \( t \)
- \( S_t \) = price of saplings in year \( t \)
- \( Y_t \) = per capita income in the metro area in year \( t \) (in thousands of dollars)
- \( H_t \) = average housing price in the metro area in year \( t \) (in thousands of dollars)

### Regression Coefficients and Standard Errors:
- Intercept: 450 (standard error = 17)
- \( P_t \): -5 (standard error = 0.13)
- \( S_t \): -225 (standard error = 0.16)
- \( Y_t \): 3 (standard error = 0.08)
- \( H_t \): 12 (standard error = 6.22)

### Step 1: Calculate t-Statistics
The t-statistic for each coefficient is calculated using the formula:

\[
t = \frac{\text{coefficient}}{\text{standard error}}
\]

#### t-Statistics Calculations:
1. **Intercept:**
   \[
   t = \frac{450}{17} \approx 26.47
   \]

2. **Price of Trees (\( P_t \)):**
   \[
   t = \frac{-5}{0.13} \approx -38.46
   \]

3. **Price of Saplings (\( S_t \)):**
   \[
   t = \frac{-225}{0.16} \approx -1406.25
   \]

4. **Per Capita Income (\( Y_t \)):**
   \[
   t = \frac{3}{0.08} \approx 37.50
   \]

5. **Housing Prices (\( H_t \)):**
   \[
   t = \frac{12}{6.22} \approx 1.93
   \]

### Step 2: Interpretation of t-Statistics
- **Intercept (t ≈ 26.47):** This t-statistic is very high, suggesting the intercept is statistically significant.
- **Price of Trees (\( P_t \)) (t ≈ -38.46):** This t-statistic is large in magnitude and negative, indicating a highly significant negative relationship between tree price and demand.
- **Price of Saplings (\( S_t \)) (t ≈ -1406.25):** The extremely large negative t-statistic implies a very significant negative effect of sapling prices on tree demand.
- **Per Capita Income (\( Y_t \)) (t ≈ 37.50):** This t-statistic indicates a strong positive significance of income on tree demand.
- **Housing Prices (\( H_t \)) (t ≈ 1.93):** This t-statistic is relatively low, indicating that the effect of housing prices on tree demand is marginally significant or potentially insignificant at common significance levels (e.g., 5%).

### Step 3: R-squared Interpretation
- **R² = 0.844:** This indicates that approximately 84.4% of the variation in tree demand is explained by the model. This is a high R² value, suggesting the model has good explanatory power.

### Summary of Evaluation:
- The model's coefficients are mostly highly significant, with very high t-statistics for most variables, except for housing prices which has a t-statistic close to 2.
- The negative coefficients for \( P_t \) and \( S_t \) suggest that as the prices of trees and saplings increase, the demand for full-sized trees decreases.
- The positive coefficients for \( Y_t \) and \( H_t \) suggest that higher per capita income and housing prices are associated with increased demand for full-sized trees.
- The model explains a large portion of the variability in tree demand (84.4%).

Do you have any questions or would you like more details on any part of this evaluation?

Here are 5 related questions to consider:
1. How would the demand model change if the R² value was significantly lower?
2. What could be the reason for the marginal significance of housing prices on tree demand?
3. How could multicollinearity affect the results in this regression model?
4. What implications do these results have for setting the price of full-sized trees?
5. How would you interpret the intercept in the context of tree demand?

**Tip:** When evaluating the significance of regression coefficients, it's essential to compare the t-statistics to critical values based on the desired significance level (e.g., 1.96 for a 5% significance level with a large sample size).

Garden City Plants is planning its full-sized tree purchases for the next several years, and is using the model of tree demand developed from past experience. The manager is responsible for developing the tree demand model. Garden City Plants is the leader in the community, and other suppliers follow its plans. The model the manager uses is the following, estimated on 34 years of past data. Qt = 450 - 5Pt - 225St + 3Yt + 12Ht, R² = 0.844, (standard errors in parentheses) where Qt = full-sized tree demand in year t, Pt = price of trees in year t, St = price of saplings in year t, Yt = per capita income in the metro area in year t, in thousands of dollars, and Ht = average housing price in the metro area in year t, in thousands of dollars. a. Evaluate these regression results, including computation of t-statistics, coefficients, and R².

Explore the regression analysis of the tree demand model for Garden City Plants, including the evaluation of coefficients, t-statistics, and R². This detailed guide is ideal for undergraduate students studying econometrics or statistics.

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