The demand function for good X is Qd x a � bPx � cM � e, where Px is the
price of good X and M is income. Least squares regression reveals that â
8.27, bˆ �2.14, cˆ 0.36, s
aˆ 5.32, sbˆ 0.41, and scˆ 0.22. The
R-squared is 0.35.
a. Compute the t-statistic for each of the estimated coefficients.
b. Determine which (if any) of the estimated coefficients are statistically different from zero.
c. Explain, in plain words, what the R-square in this regression indicates.
a. Sure, let me compute those t-statistics for you.
The t-statistic is given by the formula: t = (estimated coefficient - hypothesized value) / standard error of the estimated coefficient.
For the coefficient a: t_a = (8.27 - 0) / 5.32 = 1.55
For the coefficient b: t_b = (2.14 - 0) / 0.41 = 5.22
For the coefficient c: t_c = (0.36 - 0) / 0.22 = 1.64
b. To determine which coefficients are statistically different from zero, we compare the t-statistic with a critical value from the t-distribution. The critical value depends on the degrees of freedom and the desired level of significance.
Now, assuming a 95% level of significance and using a two-tailed test (since we are interested in both positive and negative deviations from zero), the critical t-value is approximately 2.052.
Comparing the t-statistics with the critical value:
- For coefficient a, 1.55 < 2.052, so it is not statistically different from zero.
- For coefficient b, 5.22 > 2.052, so it is statistically different from zero.
- For coefficient c, 1.64 < 2.052, so it is not statistically different from zero.
c. The R-squared value in this regression (0.35) indicates the proportion of the variation in the quantity demanded of good X that is explained by the independent variables (i.e., price and income). In simple terms, it tells us how well the regression equation fits the data. In this case, approximately 35% of the variation in the quantity demanded is explained by the price and income variables. The remaining 65% could be attributed to other factors not included in the model, such as taste preferences or competitor prices.
a. To compute the t-statistic for each estimated coefficient, we use the formula:
t = (estimate - hypothesized value) / standard error
For the coefficient a:
t_a = (8.27 - 0) / 5.32 ≈ 1.55
For the coefficient b:
t_b = (2.14 - 0) / 0.41 ≈ 5.22
For the coefficient c:
t_c = (0.36 - 0) / 0.22 ≈ 1.64
b. To determine which estimated coefficients are statistically different from zero, we compare the t-statistic values to the critical t-value at a given level of significance. If the absolute value of the t-statistic is greater than the critical t-value, then the coefficient is statistically different from zero. The critical t-value depends on the level of significance and the degrees of freedom. Let's assume a 5% level of significance (α = 0.05).
For α = 0.05 and degrees of freedom (n - k - 1 = 0, where n is the number of observations and k is the number of coefficients), we can consult a t-table or use statistical software to find the critical t-value.
c. The R-squared in this regression indicates the proportion of the variation in the quantity demanded of good X that is explained by the independent variables (price and income) included in the demand function. In this case, an R-squared of 0.35 means that approximately 35% of the variation in quantity demanded can be explained by price and income changes, while the remaining 65% is attributed to other factors not included in the model. A higher R-squared would indicate a stronger relationship between the independent variables and the dependent variable, suggesting that the demand function provides a better explanation for changes in quantity demanded.
a. To compute the t-statistic for each estimated coefficient, we need to use the formula:
t = (estimated coefficient - hypothesized value) / standard error of the estimated coefficient
For the given coefficients, let's calculate the t-statistic:
For coefficient a: t = (8.27 - 0) / 5.32 = 1.557
For coefficient b: t = (2.14 - 0) / 0.41 = 5.220
For coefficient c: t = (0.36 - 0) / 0.22 = 1.636
b. To determine if the estimated coefficients are statistically different from zero, we compare the absolute value of the calculated t-statistic with the critical value for a given significance level (usually 5% or 1%). If the absolute value of the t-statistic is greater than the critical value, then the coefficient is considered statistically different from zero.
Based on the t-statistics calculated in part (a), we can compare them to the critical values and determine for each coefficient:
For coefficient a: The critical value depends on the desired significance level, degrees of freedom, and considering a two-tailed test (e.g., 1.96 for a 5% significance level). Since the absolute value of the t-statistic (1.557) is smaller than the critical value, we cannot consider coefficient a statistically different from zero.
For coefficient b: Using the same approach, the critical value at a 5% significance level is 1.96. Since the absolute value of the t-statistic (5.220) is greater than the critical value, we can consider coefficient b statistically different from zero.
For coefficient c: Using the same approach, the absolute value of the t-statistic (1.636) is smaller than the critical value, so we cannot consider coefficient c statistically different from zero.
c. The R-squared (R²) in this regression indicates the proportion of the total variation in the dependent variable (Qd) that can be explained by the independent variables (Px and M) included in the regression model. In other words, it measures the goodness-of-fit of the regression model.
In this case, an R-squared of 0.35 means that 35% of the variation in the demand for good X (Qd) can be explained by the variation in its price (Px) and income (M), as captured by the regression equation. The remaining 65% of the variation is unaccounted for and could be due to other factors not included in the model or random fluctuations. Overall, a higher R-squared indicates a better fit between the model and the actual data, indicating that the independent variables have a stronger influence on the dependent variable.