Consider the regression model Y; = B1X1¡ + B2X2; + B3 (X1i * X2;) + Uj. Show that (a) AY/AX1 = B1 + B3X2 (effect of a change in X1 holding X2 constant). (b) AY/AX2 = ß1 + B3X1 (effect of a change in X2 holding X1 constant). (c) If X1 changes AX1 and X2 changes AX2, then AY = (B1 + B3X2) AX1 + (B2 + B3X1)AX2+ B3AX1AX2.
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- 1. Consider a linear regression model y = XB + € with E(e) = 0. The bias of the ridge estimator of 3 obtained by minimizing Q(B) = (y — Xß)¹ (y — Xß) + r(BTB), for some r > 0, is ——(X²X + r1)-¹8 1 (X¹X +rI)-¹3 r -r(XTX+rI) ¹8 r(X¹X+r1) ¹3The table to the right contains price-demand and total cost data for the production of projectors, where p is the wholesale price (in dollars) of a projector for an annual demand of x projectors and C is the total cost (in dollars) of producing x projectors. Answer the following questions (A) - (D). (A) Find a quadratic regression equation for the price-demand data, using x as the independent variable. X 270 360 520 780 The fixed costs are $. (Round to the nearest dollar as needed.) ITTI y = (Type an expression using x as the variable. Use integers or decimals for any numbers in the expression. Round to two decimal places as needed.) Use the linear regression equation found in the previous step to estimate the fixed costs and variable costs per projector. The variable costs are $ per projector. (Round to the nearest dollar as needed.) (C) Find the break even points. The break even points are (Type ordered pairs. Use a comma to separate answers as needed. Round to the nearest integer as…1. You are interested the causal effect of X on Y, B1. Suppose that X, and X2 are uncorrelated. You estimate B1 by regressing Y onto X1 (so that X2 is not included in the regression). Does this estimator suffer from omitted variable bias due to the exclusion of X2? (a) Yes (b) No (c) Maybe 2. Omitted variable bias violates which of the following assumptions: (a) The conditional distribution of u, given X1i X2i, ...Xki has a mean of zero (b) (Xi, X2i...Y;), i = 1, ., n are independently and identically distributed (c) Heteroskedasticity (d) Perfect multicollinearity
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- Styles 6. The following two regression models are Probit and Logit respectively: Pr(Y=1|X)=(Bo+Bi xXr+Bzx X2) Pr(Y=1|X)=F(Bu + B₁ × Xs + B₂ × X2) (a) what functions do and F represent? (b) How do Probit, and Logit ensure that the predicted probabilities are always between 0 and 1? (c) Sketch a graph of the Y= $(Z) function. (Z on the horizontal axis, Y on the vertical axis) (d) What estimation method is used to estimate the coefficients in a Probit/Logit model? (e) What are the two measures of fit for models with binary dependent variables? Focus 88 B E2 In the simple linear regression model y = Bo + Bjx + u, suppose that E(u) + 0. Letting a, = E(u), show that the model can always be rewritten with the same slope, but a new intercept and error, where the new error has a zero expected value. 3 The following table contains the ACT scores and the GPA (grade point average) for eight college stu- dents. Grade point average is based on a four-point scale and has been rounded to one digit after the decimal. Student GPA ACT 1 2.8 21 2 3.4 24 3 3.0 26 4 3.5 27 3.6 29 6 3.0 25 7 2.7 25 3.7 30 (i) Estimate the relationship between GPA and ACT using OLS; that is, obtain the intercept and slope estimates in the equation GPA = B, + B¡ACT. Comment on the direction of the relationship. Does the intercept have a useful interpretation here? Explain. How much higher is the GPA predicted to be if the ACT score is increased by five points? (ii) Compute the fitted values and residuals for each observation, and verify that the residuals (approximately) sum…QUESTION 1 In the equation, y = 8o + Bjx1 + 8zx2 + u, 8z is a(n) O a. intercept parameter O b. slope parameter O. dependent variable O d. independent variable QUESTION 2 If an independent variable in a multiple linear regression model is an exact linear combination of other independent variables, the model suffers from the problem of O a. perfect collinearity O b.heteroskedasticty O . homoskedasticity O d. omitted variable bias QUESTION 3 Which of the following is true of R 2? O a. R- usually decreases with an increase in the number of independent variables in a regression. O b.R2 shows what percentage of the total variation in the dependent variable, Y, is explained by the explanatory variables. OC.A low R2 indicates that the Ordinary Least Squares line fits the data well. O d. R² is also called the standard error of regression. QUESTION 4 We estimate the model Wage, = -2.91+0.568educ; + 0.033 exper; +0.115 tenure; by OLS, where wage is the hourly wage of a worker measured in dollars,…
- (Econmetrics) Q.1 How can you test for general misspecification of model if it would have only (any of) two independent variables?2. Consider a two variable regression model, which satisfies all the Gauss Markov assumptions except that the error variance is proportional to X² i.e.E(u?) = o²X? Y₁ = B₁ + B₂X₁ + Ui How would you obtain the best linear unbiased estimates from the above regression.2. In a multiple regression of y on x1, x2, and x3, including additional variables on the right-hand side of the model always increases R2. (True/False).