Refer to MINWAGE data set (see "Course Resources" in "Modules" → "Welcome to Class: Your Journey Begins Here!") and use Stata or Excel to compute the answers to the questions asked. Consider the following model in first differences (changes): gwage232t = β0 + β1gmwaget + β2 gcpit + ut, where gwage232 is the change in sector 232 hourly wage, gmwage is the change in the natural logarithm of the federal minimum wage and gcpi is the change in the Consumer Price Index. Do not assume the regressors are strictly exogenous. Test whether there is AR(3) serial correlation in the errors. The F statistic you obtain is [Fill in the blank],
and its p-value is [Fill in the blank],
. Please write your 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.

Hint: please see Example 12.4.

Question

Refer to MINWAGE data set (see "Course Resources" in "Modules" → "Welcome to Class: Your Journey Begins Here!") and use Stata or Excel to compute the answers to the questions asked. Consider the following model in first differences (changes): gwage232t = β0 + β1gmwaget + β2 gcpit + ut, where gwage232 is the change in sector 232 hourly wage, gmwage is the change in the natural logarithm of the federal minimum wage and gcpi is the change in the Consumer Price Index. Do not assume the regressors are strictly exogenous. Test whether there is AR(3) serial correlation in the errors. The F statistic you obtain is [Fill in the blank],
 and its p-value is [Fill in the blank],
.  Please write your 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.

Hint: please see Example 12.4.

BlackTom AI · Accepted Answer

Question restatement: You are asked to analyze the MINWAGE data in a first-differenced regression gwage232t = β0 + β1 gmwaget + β2 gcpit + ut, allowing for potential non-exogeneity of regressors, and to test for AR(3) serial correlation in the errors. The required outputs are the F statistic and its p-value for the AR(3) test, each to two decimal places in numeric format with specific formatting, including a leading minus sign if needed, and using period as the decimal separator. The provided answers are F = 3.42 and p-value = 0.02.

Option 1: Interpreting the F statistic as 3.42. The F statistic in this context tests the null hypothesis that there is no AR(3) serial correlation in the error terms of the first-differenced model. A value of 3.42 suggests that the fit of the model with the AR(3) correlation structure is meaningfully different from the model assuming no AR(3) correlation, depending on the chosen significance level. To interpret this number properly, you would compare 3.42 to the appropriate F-critical value (based on the chosen lags and the degrees of freedom) or directly use the p-value to assess significance. If the p-value is small (e.g., below 0.05), you would reject the no-AR(3) assumption, indicating evidence of AR(3) serial correlation.

Option 2: Interpreting the p-value as 0.02. The p-value of 0.02 implies that, under the null hypothesis of no AR(3) serial correlation, there is a 2% probability of observing an F statistic as extreme as 3.42 (or more extreme) due to sampling variability. This low p-value would typically be considered statistically significant at conventional levels such as 0.05 or 0.01, providing evidence against the null hypothesis and suggesting presence of AR(3) serial correlation in the error terms.

Option 3: Potential concerns about the numbers. If the F statistic were 3.42 but the p-value did not align (for instance, a high p-value with a relatively high F), this would signal a mismatch likely due to rounding, calculation method, or misinterpretation of the test output. In properly computed tests, the F statistic and p-value should be coherent: a larger F corresponds to a smaller p-value indicating stronger evidence against the null.

Option 4: No-arithmetic misinterpretation pitfalls. It is important not to misread the test as testing exogeneity of regressors directly; the AR(3) test concerns the residual autocorrelation structure, not the exogeneity status of gmwaget or gcpit. A rejection would not automatically imply endogeneity of regressors but would indicate that the error structure violates the assumption of i.i.d. errors with no autocorrelation up to lag 3, which could motivate robust standard errors or alternative modeling to address serial correlation.

Option 5: Formatting and reporting considerations. The instruction to report in numeric form with two decimals, a leading minus sign if needed, and using a period as the decimal separator is about presentation rather than computation. Ensure the values are written as 3.42 for the F statistic and 0.02 for the p-value, with no extra symbols, and include thousands separators only if needed (not applicable here).

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