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4. (12 pts) Suppose that (Y, X:) satisfy the three assumptions we made in the regression...
Based on Stock & Watson "Introduction to Econometrics" 6th ed., Exercise 5.8.
(Based on Stock & Watson "Introduction to Econometrics" 6th ed., Exercise 5.8.) Suppose that \(\left(Y_{i}, X_{i}\right)\) satisfy the simple linear regression assumptions. In addition, \(u_{i}\) is \(N\left(0, \sigma_{u}^{2}\right)\) and is independent of \(X_{i}\). A sample of size \(n=30\) yields$$ \hat{Y}_{i}=43.2+61.5 X_{i}, n=30, R^{2}=0.54 . $$$$ (10.2)(7.4) $$(a) Construct a \(95 \%\) confidence in(a) Construct a \(95 \%\) confidence interval for \(\beta_{0}\).(b) Test \(H_{0}: \beta_{1}=55\) v.s. \(H_{1}: \beta_{1} \neq 55\) at the significance level \(5 \%\).(c) Test \(H_{0}: \beta_{1}=55\) v.s. \(H_{1}:...
Suppose that (Yį, X;) satisfy the assumptions SLR.1 - SLR.4. A random sample of size n...
Suppose that (Yį, X;) satisfy the assumptions SLR.1 - SLR.4. A random sample of size n = 250 is drawn and yields Y = 5.4+3.2X, n = 250, R2 = 0.26. (3.1) (1.5) (a) Test Ho: B1 = 0 against H :B1 + 0 at the significance level 5%. (b) Construct a 95% confidence interval for B1. (c) Suppose you learned that Y; and X; were independent. Would you be surprised? Explain. (d) Suppose thet Y; and X, are independent...
lab 1? Suppose that (Y, X) staly the assumptions specitled here and in addition, , s...
lab 1? Suppose that (Y, X) staly the assumptions specitled here and in addition, , s (0,0%) and independent of X, A random sample of a 14 in draw and yoide Question 9-4123 + 73 70x, R-012, SER. 16 (14.9) (5.8) Q: 100=0.17 of capita Where the numbers in parentheses are the homoskedastic-only standard errors for the regression coefficients, and respectively. Refer to the student distribution Subscrip with n - 2 degrees of freedom to answer the following questions A:...
2. Changing Units Suppose we estimate a standard OLS regression equation on data X and Y...
2. Changing Units Suppose we estimate a standard OLS regression equation on data X and Y and have the standard formulas: 3, L2=(X - X)(Y - Ý) 1=1(X; - X)2 Bo =Ỹ – BX Now suppose that Xi = 1+1.6Z; for some Zi. I.e., suppose that X; were generated by trans- forming some Zi. a) Show that: X = do +ajz for some do, ai (that you need to solve for). b) Plug in the expression from (a) into (X;...
2. Suppose we observe the pairs (X, Y), i-1, , n and fit the simple linear regression (SLR) model...
2. Suppose we observe the pairs (X, Y), i-1, , n and fit the simple linear regression (SLR) model Consider the test H0 : β,-0 vs. Ha : Aメ0. (a) What is the full model? Write the appropriate matrices Y and X. (b) What is the full model SSE? (c) What is the reduced model? Write the appropriate matrix XR. (d) What is the reduced model SSE? (e) Simplify the F statistics of the ANOVA test of Ho B10 vs....
4. Let X,X,Bernoulli(p), and let Y Xi. Then we know that Y-Binomial(n.p) Consider the hypotheses Hop-Po...
4. Let X,X,Bernoulli(p), and let Y Xi. Then we know that Y-Binomial(n.p) Consider the hypotheses Hop-Po against Hip#po- a. c. For the particular case of po0.25 and n 5, fill in the table: 0 3 4 λ(y) P(Y - y) Find the (generalized) likelihood ratio test φ(y) of size α for testing Ho:p-po vs. H,: p d. po. Your test should be expressed in terms of y and α.
4. Exercise Let X, Y be RVs. Denote E[X] = Hy and E[Y] =py. Suppose we...
4. Exercise Let X, Y be RVs. Denote E[X] = Hy and E[Y] =py. Suppose we want to test the null hypothesis Ho : Mx = uy against the alternative hypothesis Hi : 4x > uy. Suppose we have i.i.d. pairs (X1,Yı),...,(Xn, Yn) from the joint distribution of (X,Y). Further assume that we know the X - Y follows a normal distribution. (i) Show that exactly) T:= (X-Y)-(ux-uy) - tn-1), Sin (3) where s2 = n-1 [?-,((X; – Y;) –...
4. (24 marks) Suppose that the random variables Yi,..., Yn satisfy Y-B BX,+ Ei, 1-1, , n, where βο and βι are parameters, X1, ,X, are con- stants, and e1,... ,en are independent and identically distr...
4. (24 marks) Suppose that the random variables Yi,..., Yn satisfy Y-B BX,+ Ei, 1-1, , n, where βο and βι are parameters, X1, ,X, are con- stants, and e1,... ,en are independent and identically distributed ran- dom variables with Ei ~ N (0,02), where σ2 is a third unknown pa- rameter. This is the familiar form for a simple linear regression model, where the parameters A, β, and σ2 explain the relationship between a dependent (or response) variable Y...
FR2 (4+4+4 12 points) (a) Let XI, X2, X10 be a randoin sample from N(μι,σ?) and Yi, Y2, 10 , Y 15 be a random sample from N (μ2, σ2), where all parameters are unknown. Sup- pose Σ 1 (Xi X 2 0 321 (Y-...
FR2 (4+4+4 12 points) (a) Let XI, X2, X10 be a randoin sample from N(μι,σ?) and Yi, Y2, 10 , Y 15 be a random sample from N (μ2, σ2), where all parameters are unknown. Sup- pose Σ 1 (Xi X 2 0 321 (Y-Y )2-100. obtain a 99% confidence interval for σ of having the form b, 0o) for some number b (No derivation needed). (b) 60 random points are selected from the unit interval (r:0 . We want...
Need help with stats true or false questions Decide (with short explanations) whether the following statements are true or false a) We consider the model y-Ao +A(z) +E. Let (-0.01, 1.5) be a 95% con...
Need help with stats true or false questions
Decide (with short explanations) whether the following statements are true or false a) We consider the model y-Ao +A(z) +E. Let (-0.01, 1.5) be a 95% confidence interval for A In this case, a t-test with significance level 1% rejects the null hypothesis Ho : A-0 against a two sided alternative. b) Complicated models with a lot of parameters are better for prediction then simple models with just a few parameters c)...
Suppose that (Yį, X;) satisfy the assumptions SLR.1 - SLR.4. A random sample of size n = 250 is drawn and yields Y = 5.4+3.2X, n = 250, R2 = 0.26. (3.1) (1.5) (a) Test Ho: B1 = 0 against H :B1 + 0 at the significance level 5%. (b) Construct a 95% confidence interval for B1. (c) Suppose you learned that Y; and X; were independent. Would you be surprised? Explain. (d) Suppose thet Y; and X, are independent...
lab 1? Suppose that (Y, X) staly the assumptions specitled here and in addition, , s (0,0%) and independent of X, A random sample of a 14 in draw and yoide Question 9-4123 + 73 70x, R-012, SER. 16 (14.9) (5.8) Q: 100=0.17 of capita Where the numbers in parentheses are the homoskedastic-only standard errors for the regression coefficients, and respectively. Refer to the student distribution Subscrip with n - 2 degrees of freedom to answer the following questions A:...
2. Changing Units Suppose we estimate a standard OLS regression equation on data X and Y and have the standard formulas: 3, L2=(X - X)(Y - Ý) 1=1(X; - X)2 Bo =Ỹ – BX Now suppose that Xi = 1+1.6Z; for some Zi. I.e., suppose that X; were generated by trans- forming some Zi. a) Show that: X = do +ajz for some do, ai (that you need to solve for). b) Plug in the expression from (a) into (X;...
2. Suppose we observe the pairs (X, Y), i-1, , n and fit the simple linear regression (SLR) model Consider the test H0 : β,-0 vs. Ha : Aメ0. (a) What is the full model? Write the appropriate matrices Y and X. (b) What is the full model SSE? (c) What is the reduced model? Write the appropriate matrix XR. (d) What is the reduced model SSE? (e) Simplify the F statistics of the ANOVA test of Ho B10 vs....
4. Let X,X,Bernoulli(p), and let Y Xi. Then we know that Y-Binomial(n.p) Consider the hypotheses Hop-Po against Hip#po- a. c. For the particular case of po0.25 and n 5, fill in the table: 0 3 4 λ(y) P(Y - y) Find the (generalized) likelihood ratio test φ(y) of size α for testing Ho:p-po vs. H,: p d. po. Your test should be expressed in terms of y and α.
4. Exercise Let X, Y be RVs. Denote E[X] = Hy and E[Y] =py. Suppose we want to test the null hypothesis Ho : Mx = uy against the alternative hypothesis Hi : 4x > uy. Suppose we have i.i.d. pairs (X1,Yı),...,(Xn, Yn) from the joint distribution of (X,Y). Further assume that we know the X - Y follows a normal distribution. (i) Show that exactly) T:= (X-Y)-(ux-uy) - tn-1), Sin (3) where s2 = n-1 [?-,((X; – Y;) –...
4. (24 marks) Suppose that the random variables Yi,..., Yn satisfy Y-B BX,+ Ei, 1-1, , n, where βο and βι are parameters, X1, ,X, are con- stants, and e1,... ,en are independent and identically distributed ran- dom variables with Ei ~ N (0,02), where σ2 is a third unknown pa- rameter. This is the familiar form for a simple linear regression model, where the parameters A, β, and σ2 explain the relationship between a dependent (or response) variable Y...
FR2 (4+4+4 12 points) (a) Let XI, X2, X10 be a randoin sample from N(μι,σ?) and Yi, Y2, 10 , Y 15 be a random sample from N (μ2, σ2), where all parameters are unknown. Sup- pose Σ 1 (Xi X 2 0 321 (Y-Y )2-100. obtain a 99% confidence interval for σ of having the form b, 0o) for some number b (No derivation needed). (b) 60 random points are selected from the unit interval (r:0 . We want...
Need help with stats true or false questions
Decide (with short explanations) whether the following statements are true or false a) We consider the model y-Ao +A(z) +E. Let (-0.01, 1.5) be a 95% confidence interval for A In this case, a t-test with significance level 1% rejects the null hypothesis Ho : A-0 against a two sided alternative. b) Complicated models with a lot of parameters are better for prediction then simple models with just a few parameters c)...
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