Question

In Section \(4-5 .\) we used as an example testing the rationality of assessments of housing prices. There, we used a log-log model in price and assess [see equation ( \(4.47)\) ]. Here, we use a level-level formulation. i. In the simple regression model $$\text { price }=\beta_{0}+\beta_{1} \text { assess }+u$$ the assessment is rational if \(\beta_{1}=1\) and \(\beta_{0}=0 .\) The estimated equation is $$\begin{aligned} \widehat{\text {price}} &=-14.47+.976 \text { assess} \\ &(16.27)(.049) \\ n &=88, \mathrm{SSR}=165,644.51, R^{2}=.820 \end{aligned}$$ First, test the hypothesis that \(\mathrm{H}_{0}: \beta_{0}=0\) against the two- sided alternative. Then, test \(\mathrm{H}_{0}: \beta_{1}=1\) against the two- sided alternative. What do you conclude? ii. To test the joint hypothesis that \(\beta_{0}=0\) and \(\beta_{1}=1\), we need the SSR in the restricted model. This amounts to computing \(\sum_{i=1}^{n}\left(\text {price}_{i}-\text {assess}_{i}\right)^{2},\) where \(n=88,\) because the residuals in the restricted model are just price \(_{i}\) - assess \(_{i}\). (No estimation is needed for the restricted model because both parameters are specified under \(\mathrm{H}_{0}\).) This turns out to yield \(\mathrm{SSR}=209,448.99\) Carry out the \(F\) test for the joint hypothesis. iii. Now, test \(\mathrm{H}_{0}: \beta_{2}=0, \beta_{3}=0,\) and \(\beta_{4}=0\) in the model $$\text { price }=\beta_{0}+\beta_{1} \text { assess }+\beta_{2} \text { lotsize }+\beta_{3} \text { sqr } f t+\beta_{4} b d r m s+u$$ The \(R\) -squared from estimating this model using the same 88 houses is .829 iv. If the variance of price changes with assess, lotsize, sqrft, or bdrms, what can you say about the F test from part (iii)?

Short answer

Fail to reject both \( \beta_0 = 0 \) and \( \beta_1 = 1 \); reject joint test \( \beta_0 = 0, \beta_1 = 1 \). F-test for \( \beta_2 = \beta_3 = \beta_4 = 0 \) shows marginal evidence against \( H_0 \). Heteroscedasticity affects reliability.

Step-by-step solution

Test Hypothesis for β₀ = 0

We start by testing the null hypothesis \( H_{0}: \beta_{0} = 0 \) against the alternative hypothesis \( H_{a}: \beta_{0} eq 0 \). Given the estimated intercept \( \widehat{\beta_{0}} = -14.47 \) with a standard error of 16.27, we calculate the t-statistic:\[ t = \frac{-14.47 - 0}{16.27} \approx -0.89 \]To evaluate this, we compare it to the critical value from the t-distribution with \( n - 2 = 86 \) degrees of freedom. Generally, if \( |t| \gt 1.99 \) or another similar critical value depending on the significance level, we reject \( H_0 \). However, since \( |t| = 0.89 \) is likely less than the critical value, we fail to reject \( H_0 \). This suggests the intercept may statistically equal zero.

Test Hypothesis for β₁ = 1

Next, we test the null hypothesis \( H_{0}: \beta_{1} = 1 \) against the alternative hypothesis \( H_{a}: \beta_{1} eq 1 \). Here, the estimated slope is \( \widehat{\beta_{1}} = 0.976 \) with a standard error of 0.049. Calculate the t-statistic:\[ t = \frac{0.976 - 1}{0.049} \approx -0.49 \]We compare this to the critical value from the t-distribution with 86 degrees of freedom. Since \( |t| = 0.49 \) is likely less than the critical value, we fail to reject \( H_{0} \). The slope is not statistically different from one.

F-Test for Joint Hypothesis β₀ = 0 and β₁ = 1

We test the joint hypothesis \( H_{0}: \beta_{0} = 0 \) and \( \beta_{1} = 1 \). The given sum of squared residuals (SSR) for the original model is 165,644.51, and for the restricted model, it is 209,448.99. The F-statistic is computed as follows:\[F = \frac{(SSR_{r} - SSR_{ur}) / q}{SSR_{ur} / (n-k-1)}\]where SSR\(_{r}\) is for the restricted model, SSR\(_{ur}\) for the unrestricted model, \( q = 2 \) (the number of restrictions), \( n = 88 \), and \( k = 1 \) (number of explanatory variables in the unrestricted model).\[F = \frac{(209,448.99 - 165,644.51) / 2}{165,644.51 / (88 - 1 - 1)} \approx 11.23\]Compare this to the critical value of the F-distribution with \( q = 2 \) and \( n - k - 1 = 86 \) degrees of freedom. If \( F \gg 3.10 \) at a 5% significance level, we reject \( H_{0} \) and conclude that the joint hypothesis is not satisfied.

Test Hypothesis for β₂ = 0, β₃ = 0, β₄ = 0

To test \( H_{0}: \beta_{2} = 0, \beta_{3} = 0, \beta_{4} = 0 \) in the expanded model with additional variables: \( \text{lotsize}, \text{sqrft}, \text{bdrms} \), use:\[F = \frac{(R_{ur}^2 - R_{r}^2) / q }{(1 - R_{ur}^2) / (n - k - 1)}\]Incremental \( R^2 \) is from .820 (restricted) to .829 (unrestricted), with \( q = 3 \).\[F = \frac{(.829 - .820) / 3}{(1 - .829) / (88 - 5)} \approx 2.25\]Evaluate against F-critical value with \( q = 3 \) and \( n - k - 1 = 83 \). If \( F \text{ is significantly larger than the critical value,}\) reject \( H_0 \).

Discuss Effects of Heteroscedasticity on F-Test

If the variance of price changes with the independent variables, this indicates heteroscedasticity. Heteroscedasticity can inflate or deflate the test statistics, affecting the reliability of the F-test. Standard errors won't be consistent, leading to unreliable conclusions where the nominal significance levels may not achieve actual significance.

Key concepts

Hypothesis Testing

Hypothesis testing is a fundamental concept in econometrics. It helps us determine if our assumptions about population parameters are true or not. In regression analysis, hypotheses often pertain to the coefficients of the model.

We start with a null hypothesis, denoted as \( H_0 \), which usually represents a statement of no effect or no difference. This is tested against an alternative hypothesis, \( H_a \). For instance, in the context of housing prices, we might hypothesize that the regression coefficient \( \beta_1 \) equals 1. This would mean the assessed value and the actual price have a one-to-one correspondence.

A t-test is commonly used to test hypotheses about individual coefficients. The test statistic is calculated, and its value is compared to a critical value from the t-distribution. If the absolute value of the calculated statistic exceeds this critical value, we reject the null hypothesis.

An important aspect of hypothesis testing is ensuring that conclusions are drawn with an appropriate level of statistical confidence, typically set at 5% or 1% significance levels.

Regression Analysis

Regression analysis is a statistical technique used to determine the relationship between variables. It is crucial for predicting and forecasting trends. In essence, it involves fitting a line or curve to a set of data points.

In our housing price example, we are using a simple regression model to predict price based on assessments. The basic equation, \( \text{price} = \beta_0 + \beta_1 \text{assess} + u \), consists of:

• \( \beta_0 \): the intercept, representing the starting value of price when assessment score is zero.
• \( \beta_1 \): the slope, which shows the change in price for a one-unit change in assessment.
• \( u \): the error term that captures unexplained variation.

By estimating \( \beta_0 \) and \( \beta_1 \) using data, we can assess the strength and type of relationship between variables. The goodness-of-fit can be evaluated using \( R^2 \), which indicates how much of the variation in the dependent variable (price) is explained by the model.

F-test

The F-test is a statistical test used to compare models and test multiple hypotheses. It's particularly useful in this exercise for testing joint hypotheses about regression coefficients.

For example, we might be interested in testing a joint hypothesis that \( \beta_0 = 0 \) and \( \beta_1 = 1 \). The F-statistic is obtained by comparing the sum of squared residuals (SSR) from a restricted model, which constrains these parameters, with the SSR from an unrestricted model that doesn't have these constraints.

The F-statistic formula is:\[F = \frac{(SSR_r - SSR_{ur}) / q}{SSR_{ur} / (n-k-1)}\]where:

• \(SSR_r\) is the residual sum of squares for the restricted model.
• \(SSR_{ur}\) is the residual sum of squares for the unrestricted model.
• \(q\) is the number of restrictions.
• \(n\) is the sample size.
• \(k\) is the number of explanatory variables in the unrestricted model.

A larger F-value suggests a significant difference between the models, indicating that at least one of the hypotheses under test is incorrect.

Heteroscedasticity

Heteroscedasticity refers to the presence of non-constant variance in the error terms of a regression model. This is a crucial consideration in econometric analysis because it can lead to inefficient estimates and affect hypothesis testing.

In regression analysis, we assume that the residuals (differences between observed and predicted values) should have constant variance. When this is not the case, standard errors may be skewed, leading to unreliable statistical tests.

If heteroscedasticity is present in our housing price model, it might mean the variability of price changes with variables like assessment, lotsize, or other factors. Heteroscedasticity can cause:

• Inflated or deflated F-statistics, potentially leading to incorrect conclusions.
• Bias in standard errors, affecting the accuracy of confidence intervals and hypothesis tests.

Detecting heteroscedasticity often involves graphical analysis or statistical tests like the Breusch-Pagan test. If identified, it can be addressed by using robust standard errors or transforming the data.