Refer to hprice1 dataset (see "Course Resources" in "Modules" → "Welcome to Class: Your Journey Begins Here!"). Use Excel’s regression data analysis tool or Stata to compute the answers to the questions asked. Consider the following regression model: price = β0 + β1lotsize + β2sqrft + β3bdrms + u, where price is house price in $1,000s, lotsize is size of lot in square feet, sqrft is size of house in square feet and bdrms is number of bedrooms. The heteroskedasticity-robust standard error you obtain for β^2{\widehat\beta}_2 is [Fill in the blank],
. Let α = 5%. You further estimate whether the size of house is statistically different from zero, using the heteroskedasticity-robust t statistic; your conclusion is [Fill in the blank],
.

NOTE: Write your first answer in number format, with 2 decimal places of precision level; do not write your answer as a fraction. Add a leading minus sign symbol, a leading zero and trailing zeros when needed. Use a period for the decimal separator and a comma to separate groups of thousands. For your second answer, write either YES or NO.

HINT: See Example 8.1.

Question

Refer to hprice1 dataset (see "Course Resources" in "Modules" → "Welcome to Class: Your Journey Begins Here!"). Use Excel’s regression data analysis tool or Stata to compute the answers to the questions asked. Consider the following regression model: price = β0 + β1lotsize + β2sqrft + β3bdrms + u, where price is house price in $1,000s, lotsize is size of lot in square feet, sqrft is size of house in square feet and bdrms is number of bedrooms. The heteroskedasticity-robust standard error you obtain for β^2{\widehat\beta}_2  is [Fill in the blank],
.  Let α = 5%. You further estimate whether the size of house is statistically different from zero, using the heteroskedasticity-robust t statistic; your conclusion is [Fill in the blank],
.

NOTE: Write your first answer in number format, with 2 decimal places of precision level; do not write your answer as a fraction. Add a leading minus sign symbol, a leading zero and trailing zeros when needed. Use a period for the decimal separator and a comma to separate groups of thousands. For your second answer, write either YES or NO.

HINT: See Example 8.1.

BlackTom AI · Accepted Answer

This question asks you to perform a regression-based inference using a heteroskedasticity-robust framework and then to interpret the results for two blanks.

First, restating the setup: we estimate price = β0 + β1 lotsize + β2 sqrft + β3 bdrms + u, where price is in thousands of dollars. We are instructed to use heteroskedasticity-robust standard errors (robust SE) for the coefficient on sqrft, β2. The two blank responses you must fill are: (1) the numeric value of the robust SE for β̂2, expressed to two decimals with the formatting rules given, and (2) whether the size of the house (sqrft) is statistically different from zero based on the heteroskedasticity-robust t statistic, answered as YES or NO.

Option-by-option reasoning:

- Regarding the robust standard error for β̂2 (the first blank):
  - The robust SE for β̂2 is computed from the sandwich (or robust) variance estimator, which adjusts the usual OLS variance to account for heteroskedasticity without assuming constant variance across observations. It depends on the residuals, the design matrix, and the variability of the squared residuals, all evaluated at the estimated model.
  - The exact numeric value can be influenced by several factors: the scale of the dependent variable (price in thousands of dollars), the units chosen for sqrft and lotsize, multicollinearity among regressors, the sample size, and the presence of heteroskedasticity in the data. Because the question explicitly asks for a numeric value to two decimals with a leading minus sign if needed, you would obtain this from your regression output. Without running the data, we cannot independently verify the precise two-decimal figure.
  - Conceptually, this SE is not the same as the classical (model-based) SE under homoskedasticity; it will reflect potential heteroskedasticity in the errors, often leading to a larger SE if heteroskedasticity is present, or a smaller one if the variance structure happens to be favorable for that estimator.

- Regarding the hypothesis test for β2 (the second blank: YES/NO):
  - You test H0: β2 = 0 against the alternative H1: β2 ≠ 0 using the heteroskedasticity-robust t statistic t_robust = (β̂2 − 0) / SE_robust(β̂2).
  - To decide at α = 5%, you compare |t_robust| to the critical value from the t-distribution with the appropriate degrees of freedom. In large samples, the distribution is well approximated by a standard normal, so you compare to about 1.96 for a two-sided test. In smaller samples, you would use the specific degrees-of-freedom adjustment provided by your software (which can differ between packages).
  - Your conclusion (YES or NO) depends on whether the absolute value of the robust t-statistic exceeds the critical value. If it does, you reject H0 and conclude that sqrft has a statistically significant association with price. If not, you fail to reject H0 and conclude there is not statistically significant evidence that sqrft affects price in the presence of heteroskedasticity.

Important practical notes:
- Ensure you read the output correctly: the SE for β̂2 will be labeled something like “Robust Std. Error” or “Std. Err. (Robust)” next to β̂2 in your regression results. Do not confuse it with the standard error from a homoskedastic assumption.
- Pay attention to formatting rules given in the prompt for the first answer (two decimals, leading minus if negative, thousands separators, period as decimal separator). Ensure you present the numeric value exactly as specified.
- If your dataset exhibits strong heteroskedasticity, the robust SEs may differ noticeably from the classical SEs, which can change both the magnitude of the first blank and the YES/NO decision of the second blank.
- The hint (Example 8.1) suggests following a standard procedure for reporting both the robust standard error and the robust t-test result, so align your steps with that instructional example when filling in the blanks.

In summary, the two blanks require: (1) reporting the robust standard error for β̂2 to two decimals per the specified format, and (2) stating YES if |t_robust| > critical value (or equivalently if the p-value is below 0.05) and NO otherwise, reflecting whether sqrft is statistically different from zero under heteroskedasticity-robust inference.

AS.440.606.50.SP25 M09: Assignment

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