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5. Consider the following regression: where EUX] = 0 (remember that this is a stronger assumption...
Suppose assumptions SLR.1-SLR.3 are satisfied and consider a regression model of savings (sav) on income (inc):...
Suppose assumptions SLR.1-SLR.3 are satisfied and consider a regression model of savings (sav) on income (inc): inc2 xe B1inc + u, where u = Bo sav = Suppose e is a random variable with the following properties: E(einc) 0 Var(elinc) a) Does this regression satisfy the zero conditional mean b) Does this regression satisfy the homoskedasticity assumption (SLR.5)? c) In the real world, why might the variance of savings depend on income? assumption (SLR.4)?
Consider the following simple regression model: a. Suppose that OLS assumptions 1 to 4 hold true. We know that homoskedasticity assumption is statedas: Var[UjIx] = σ2 for all i Now, suppose that...
Consider the following simple regression model: a. Suppose that OLS assumptions 1 to 4 hold true. We know that homoskedasticity assumption is statedas: Var[UjIx] = σ2 for all i Now, suppose that homoskedasticity does not hold. Mathematically, this is expressed as In other words, the subscript i in σ12 means that the conditional variance of errors for each individual i is different. Under heteroskedasticity, we can derive the expression for the variance of Var(B) as SST Where SSTx is the...
Consider the following formulations of the 1 variable regression model: Y = β0 + β1x +...
Consider the following formulations of the 1 variable regression model: Y = β0 + β1x + u and Y = α0 + α1(x − ¯x) + a a) would the estimates of β0 and α0 the same? Explicitly shows this by deriving the estimates. b) What about β1 and α1 ? c) In the regression Y = β0 +β1x+u suppose we multiply each X value by a constant, say, 2. Will it change the residuals and fitted values of Y?...
Consider the following linear regression model 1. For any X x, let Y xBU, where 3...
Consider the following linear regression model 1. For any X x, let Y xBU, where 3 E R*. 2. X is exogenous 3. The probability model is {f(u;0) is a distribution on R: Ef [U] = 0, VAR, [U] = 02,0 > 0}. 4. Sampling model: Y} anidependent sample, sequentially generated using Yi x Ui,where the U IID(0,0) are (i) Let K 0 be a given number. We wish to estimate B using least-squares subject to the constraint 6BK2. Write...
6. Consider the following regression model without an intercept: Y = B,X, +U, One possible estimator...
6. Consider the following regression model without an intercept: Y = B,X, +U, One possible estimator for this model is given by: BE ANXJ Assume that you can make all of the usual ordinary least squares assumptions about the model, including the assumption that the true model does not include an intercept. Is B, an unbiased estimator? Please prove your conclusion, being sure to state the assumptions you use. [5 points]
5) Consider the simple linear regression model N(0, o2) i = 1,...,n Let g be the...
5) Consider the simple linear regression model N(0, o2) i = 1,...,n Let g be the mean of the yi, and let â and ß be the MLES of a and B, respectively. Let yi = â-+ Bxi be the fitted values, and let e; = yi -yi be the residuals a) What is Cov(j, B) b) What is Cov(â, ß) c) Show that 1 ei = 0 d) Show that _1 x;e; = 0 e) Show that 1iei =...
Consider the following simple regression model: where the e, are independent errors with E(ed-0 and var(et)-Ơ2X?...
Consider the following simple regression model: where the e, are independent errors with E(ed-0 and var(et)-Ơ2X? a. In this case, would an ordinary least squares regression provide you with the best b. c. linear unbiased estimates? Why or why not? What is the transformed model that would give you constant error variance? Given the following data: y = (4,3,1,0,2) and x = (1,2,1,3,4) Find the generalized least squares estimates of β1 and β2 (Do this by hand! Not with excel)
3. Consider the following model to explain monthly beer consumption: where E(ujincome, price, educ, female) -0...
3. Consider the following model to explain monthly beer consumption: where E(ujincome, price, educ, female) -0 and Var(ulincome, price, educ, female) - ?"income. (a) What problems arise if OLS is used to estimate the model? (b) Write the transformed equation that has a homoskedastic error term.(4 Marks) (4 Marks)
Consider the following regression results: Describe how the response y depends on the regressor x. What...
Consider the following regression results:
Describe how the response y depends on the regressor x. What is
the formula for the regression line? What is the B0 and B1, and
what do these coefficients represent? The Residuals vs. fitted plot
is used to assess what assumption? What does the above plot tell
you about your data? (remember to round all answers to 3 decimal
places)
Call: Im(formula = y ~ X, data = d) Residuals: Min 1Q Median 3Q Max...
Suppose assumptions SLR.1-SLR.3 are satisfied and consider a regression model of savings (sav) on income (inc): inc2 xe B1inc + u, where u = Bo sav = Suppose e is a random variable with the following properties: E(einc) 0 Var(elinc) a) Does this regression satisfy the zero conditional mean b) Does this regression satisfy the homoskedasticity assumption (SLR.5)? c) In the real world, why might the variance of savings depend on income? assumption (SLR.4)?
Consider the following simple regression model: a. Suppose that OLS assumptions 1 to 4 hold true. We know that homoskedasticity assumption is statedas: Var[UjIx] = σ2 for all i Now, suppose that homoskedasticity does not hold. Mathematically, this is expressed as In other words, the subscript i in σ12 means that the conditional variance of errors for each individual i is different. Under heteroskedasticity, we can derive the expression for the variance of Var(B) as SST Where SSTx is the...
Consider the following linear regression model 1. For any X x, let Y xBU, where 3 E R*. 2. X is exogenous 3. The probability model is {f(u;0) is a distribution on R: Ef [U] = 0, VAR, [U] = 02,0 > 0}. 4. Sampling model: Y} anidependent sample, sequentially generated using Yi x Ui,where the U IID(0,0) are (i) Let K 0 be a given number. We wish to estimate B using least-squares subject to the constraint 6BK2. Write...
6. Consider the following regression model without an intercept: Y = B,X, +U, One possible estimator for this model is given by: BE ANXJ Assume that you can make all of the usual ordinary least squares assumptions about the model, including the assumption that the true model does not include an intercept. Is B, an unbiased estimator? Please prove your conclusion, being sure to state the assumptions you use. [5 points]
5) Consider the simple linear regression model N(0, o2) i = 1,...,n Let g be the mean of the yi, and let â and ß be the MLES of a and B, respectively. Let yi = â-+ Bxi be the fitted values, and let e; = yi -yi be the residuals a) What is Cov(j, B) b) What is Cov(â, ß) c) Show that 1 ei = 0 d) Show that _1 x;e; = 0 e) Show that 1iei =...
Consider the following simple regression model: where the e, are independent errors with E(ed-0 and var(et)-Ơ2X? a. In this case, would an ordinary least squares regression provide you with the best b. c. linear unbiased estimates? Why or why not? What is the transformed model that would give you constant error variance? Given the following data: y = (4,3,1,0,2) and x = (1,2,1,3,4) Find the generalized least squares estimates of β1 and β2 (Do this by hand! Not with excel)
3. Consider the following model to explain monthly beer consumption: where E(ujincome, price, educ, female) -0 and Var(ulincome, price, educ, female) - ?"income. (a) What problems arise if OLS is used to estimate the model? (b) Write the transformed equation that has a homoskedastic error term.(4 Marks) (4 Marks)
Consider the following regression results:
Describe how the response y depends on the regressor x. What is
the formula for the regression line? What is the B0 and B1, and
what do these coefficients represent? The Residuals vs. fitted plot
is used to assess what assumption? What does the above plot tell
you about your data? (remember to round all answers to 3 decimal
places)
Call: Im(formula = y ~ X, data = d) Residuals: Min 1Q Median 3Q Max...
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