Use the data set in this file. You will use the following variables:subs: number of library subscriptionsprice: library subscription pricePublisher: categorical variable with 5 possible values “Academic Press”, “Blackwell”, “Elsevier”, “Kluwer” and “Others”.pages: number of pagesAnswer the following questions.(a) Run the following regression:$Y= \beta_0 + \beta_1 X_1 + \beta_2 X_1^2 + \beta_3 X_2 +\beta_4 X_3 +\beta_5 X_4 + \beta_6 X_5 +u, $$Y$=subs, $X_1$=price, $X_2$: binary variable for Publisher (1 means “Academic Press” and 0 means “not Academic Press”) $X_3$: binary variable for Publisher (1 means “Blackwell” and 0 means “not Blackwell”) $X_4$: binary variable for Publisher (1 means “Elsevier” and 0 means “not Elsevier”) $X_5$: binary variable for Publisher (1 means “Kluwer” and 0 means “not Kluwer”) Make sure you define the binary variables. (5 points)(b) Using the regression in part (a), compute the p-value for testing $H_0: \ \beta_6=0 $ vs $H_1:\ \beta_6\neq 0 $. (5 points)(c) Using the regression in part (a), compute the p-value for testing $H_0: \ \beta_1=\beta_2=0 $ vs $H_1: $ at least one of $\beta_1, \beta_2 $ is not zero. (5 points)(d) Using the regression in part (a), compute the p-value for testing $H_0: \ \beta_5+4 \beta_6=0 $ vs $H_1:\ \beta_5 + 4 \beta_6 \neq 0 $. (Hint: use the linearHypothesis command. You can choose to transform the regression, but it might be harder than using the linearHypothesis command.) (5 points)(e) Using the regression in part (a), compute the p-value for testing $H_0: \ \beta_4+\beta_5=\beta_3+\beta_6=0 $ vs $H_1: $ at least one of $\beta_4+\beta_5$ and $ \beta_3 +\beta_6$ is not zero. (5 points)

Question

BlackTom AI · Accepted Answer

This question presents a data analysis task with several subparts (a)–(e) but does not provide explicit answer choices to evaluate. I will outline how to restate and approach each part, including definitions, hypotheses, and the statistical tests you would perform.

Part (a): Define the regression model and binary predictors
- Restate the model: Y = subs, X1 = price, X1^2 is the price squared term, and X2–X5 are binary indicators for publishers. The model is Y = β0 + β1 X1 + β2 X1^2 + β3 X2 + β4 X3 + β5 X4 + β6 X5 + u.
- Define binary variables exactly as described: create four dummy variables for publishers, with one publisher serving as the reference category (the one not included among X2–X5, which would be “Others”). For example:
  - X2 = 1 if Publisher = Academic Press, 0 otherwise.
  - X3 = 1 if Publisher = Blackwell, 0 otherwise.
  - X4 = 1 if Publisher = Elsevier, 0 otherwise.
  - X5 = 1 if Publisher = Kluwer, 0 otherwise.
  - This means the base group is Publisher = Others.
- Explain the interpretation briefly: β1 captures the linear price effect on subscriptions, β2 captures the curvature (price squared), and β3–β6 capture the incremental effect of each Publisher category relative to Others.
- Practical notes: ensure you center or scale price if necessary for numerical stability, and verify model assumptions (linearity in parameters, no perfect multicollinearity with the four dummies plus base category, homoscedasticity, normality of residuals if inference relies on t-tests, etc.).

Part (b): Test H0: β6 = 0 vs H1: β6 ≠ 0
- Identify β6: the coefficient on X5, which is the dummy for Publisher = Kluwer. The null hypothesis states that being in Kluwer has no effect on subs beyond the other terms.
- Test approach: use the t-test for the coefficient β6 (the standard error of β6 from the regression) or, equivalently, use a Wald test for H0: β6 = 0. The p-value is obtained from the t-statistic |β6| / SE(β6) with degrees of freedom equal to N − p, where p is the number of estimated parameters (including the intercept).
- If you prefer a joint assessment (not requested here), you would use an F-test, but for a single coefficient, the t-test is standard. Consider whether heteroskedasticity-robust SEs are appropriate if the default assumptions may be violated.

Part (c): Test H0: β1 = β2 = 0 vs H1: at least one of β1, β2 ≠ 0
- This is a joint hypothesis about the linear and quadratic price terms. You would perform an F-test (or a Wald test) for the joint null H0: β1 = β2 = 0.
- In practice: use a linear hypothesis test or an F-test that tests the vector [β1, β2] = [0, 0]. The test statistic compares the restricted model (excluding X1 and X1^2) to the full model, using the appropriate error degrees of freedom.
- Interpretation: rejection implies that price, price squared, or both have explanatory power for subscriptions beyond the other predictors.

Part (d): Test H0: β5 + 4 β6 = 0 vs H1: β5 + 4 β6 ≠ 0
- Here you’re testing a linear combination of two coefficients: β5 (publisher Elsevier) and β6 (publisher Kluwer). The hypothesis is about whether the combined effect equals zero.
- Use the linearHypothesis command (in R’s car package, or in Stata/SAS equivalents) or an equivalent Wald test to test Lβ = c, where L is a row vector selecting β5 and β6 and c = 0. Specifically, you would test [0, 0, 0, 0, 1, 4] · β = 0, depending on the exact parameter ordering (including β0, β1, β2, β3, β4, β5, β6).
- Alternatively, you could reparameterize or construct a new regressor representing X5 + 4*X6 (though X6 here is β6’s label; ensure you’re using the correct coefficients). The key is to test a single linear constraint on the coefficient vector.
- The p-value comes from the F-statistic (or Wald statistic) for that linear restriction.

Part (e): Test H0: β4 + β5 = β3 + β6 = 0 vs H1: at least one of the two sums is nonzero
- This is a joint hypothesis about two linear combinations of coefficients: sum1 = β4 + β5 and sum2 = β3 + β6.
- Use a multivariate linear hypothesis test (e.g., an F-test) for two restrictions simultaneously. In R, you could specify a matrix L that picks these linear forms from the coefficient vector β and test H0: Lβ = 0.
- Conceptually, you’re asking whether the combined effects of X3 and X4 (via β3 and β4) and the combined effects of X2 and X5 (via β5 and β6) are both zero. Rejection indicates that at least one of those summed effects materially influences Y.

General notes across parts
- Data preparation: verify that X2–X5 are correctly coded as mutually exclusive publisher dummies with Others as the reference group. Ensure there is no perfect multicollinearity among the regressors (which should be satisfied with exactly four dummies plus the intercept and the continuous terms).
- Inference: since multiple hypotheses are tested, consider whether to adjust for multiple comparisons or report unadjusted p-values as requested. If heteroskedasticity is suspected, use robust standard errors when computing p-values for the t-tests and F-tests.
- Software hints: in R, use lm(...) for the regression, summary() for standard errors and t-values, and the car::linearHypothesis(...) function to perform the hypotheses in parts (d) and (e). In Stata, use regress and test/ testparm commands accordingly. In Python with statsmodels, use OLS.fit(...) and f_test or t_test as appropriate.
- Ensure your interpretation remains tied to the model specification: the coefficients reflect conditional associations given price, price-squared, and publisher dummies, and the hypotheses constrain specific linear combinations of those effects.

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