Heteroscedastic data
Consider a data set in which each data point tn is associated with a weighting factor rn > 0, so that the sum-of-squares error function becomes ED(w) = 1 2 X N n=1 rn tn − w Tφ(xn) 2 . Find an expression for the solution w∗ that minimizes this error function. Give two alternative interpretations of the weighted sum-of-squares error function in terms of (i) data dependent noise variance and (ii) replicated data points. Can you see a situation where this model is usable?
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The following expression is known as the weighted average loss: N L = an(tn – w...
The following expression is known as the weighted average loss: N L = an(tn – w xn)2 n=1 where the influence of each data point is controlled by its associated a pa- rameter. Assuming that each an is fixed, derive the optimal least squares parameter value W. Hint: define matrix A be a diagonal matrix of the weights Q1, ...Qn as follows: [ai 0 ... 0 02 ... 0 | 0 0 ... an] Hint: With this matrix defined, the...
(17 points) Occasionally, a mean function in which the intercept is known to be 0 may...
(17 points) Occasionally, a mean function in which the intercept is known to be 0 may be fit. Equivalently, our simple linear regression model is Y; = B1X: + € Assume in a fixed model (z values are fixed), this mean function is given by E[Y] =B120. (a) Assuming € are i.i.d following N(0,02), the residual sum of squares for this model, is SSE = +1(Y; – B12:) Show B1, the least squares estimate of B1. Show that ſi is...
Problem 4: Complete the ANOVA table based on the following data: Replications Standard Deviation ...
Problem 4: Complete the ANOVA table based on the following data: Replications Standard Deviation Factor A Level 1 Level 2 Mean 15 ANOVA Table Degree Freedom of Sum of square Mean sum of F-valuc Source error square error Treatment Error Total NA NA NA Recall that for single-factor ANOVA, we have the following -SSTreatments+SSp where ss,-Σ Σ(y)-T)-total sum of squares n Σ(vi-r-treatment sum of squares i- SSE _ Σ Σ(w-5)2-error sum of squares Given a dataset like the following: Treatment...
All parts please. You have the results of a simple linear regression based on state-level data...
All parts please.
You have the results of a simple linear regression based on state-level data and the (a) The estimated error variance .04672. What is the sum of the squared least (b) The estimated variance of ba is 0.00098. What is the standard error of b2? What 2.7 District of Columbia, a total of N- 51 observations. squares residuals? is the value of Σ(x- 2? (c) Suppose the dependent variable y the state's mean income (in thousands of dollars)...
A) In lecture we derived the estimate of B in WLS as Derive A.wls when p-1. (It should have a for...
a) In lecture we derived the estimate of B in WLS as Derive A.wls when p-1. (It should have a form similar to simple linear regression.) (Hints: Notice that we can write a weighted average as analogues of the sums of squares identities we've used; you should derive these if you need to use them.) . You may need to use weighted b) Assume we have the following data: T1 T2 y2 That is, we have a total of n...
Problem 3 The USDA collects data on commodity prices and consumption. Use their data to explain...
Problem 3 The USDA collects data on commodity prices and consumption. Use their data to explain per-capita chicken consumption as a function of the price of chicken. (Note that economic theory suggests that an increase in a commodity price is generally associated with a decrease in demand.) 0per capita chicken consumption PCHICK consumer price index for chicken Collapse Al We estimated a linear regression model using annual data for 1950-2001 (N 52) Dependent variable: coeffidient std. error const PCHICK -0.1...
Problem 3 The USDA collects data on commodity prices and consumption. Use their data to explain...
Problem 3 The USDA collects data on commodity prices and consumption. Use their data to explain per-capita chicken consumption as a function of the price of chicken. (Note that economic theory suggests that an increase in a commodity price is generally associated with a decrease in demand.) Q capita chicken consumption PCHICKconsumer price index for chicken We estimated a linear regression model usng annual data for 1950-2001 ( N = 52 ). Dependent variable: coefficient d. error const 0.05 R-squared:...
Problem 3 The USDA collects data on commodity prices and consumption. Use their data to explain...
Problem 3 The USDA collects data on commodity prices and consumption. Use their data to explain per-capita chicken consumption as a function of the price of chicken. (Note that economic theory suggests that an increase in a commodity price is generally associated with a decrease in demand.) 0 per capita chicken consumption PCHICK consumer price index for chicken We estimated a linear regression model using annual data for 1950-2001 (N 52). Dependent variable: coefficlent std. error const РСНСК -0.1 0.15...
QUESTION 2 Let Xi.. Xn be a random sample from a N (μ, σ 2) distribution,...
QUESTION 2 Let Xi.. Xn be a random sample from a N (μ, σ 2) distribution, and let S2 and Š-n--S2 be two estimators of σ2. Given: E (S2) σ 2 and V (S2) - ya-X)2 n-l -σ (a) Determine: E S2): (l) V (S2); and (il) MSE (S) (b) Which of s2 and S2 has a larger mean square error? (c) Suppose thatnis an estimator of e based on a random sample of size n. Another equivalent definition of...
The USDA collects data on commodity prices and consumption. Use their data to explain per-capita chicken...
The USDA collects data on commodity prices and consumption. Use their data to explain per-capita chicken consumption as a function of the price of chicken. (Note that economic theory suggests that an increase in a commodity price is generally associated with a decrease in demand.) -per capita chicken consumption PCHICK consumer price index for chicken We estimated a linear regression model using annual data for 1950-2001 (N 52) Dependent variable: coefficient std. error const PCHICK -0.1 0.1 R-squared: 0.854880 a....
The following expression is known as the weighted average loss: N L = an(tn – w xn)2 n=1 where the influence of each data point is controlled by its associated a pa- rameter. Assuming that each an is fixed, derive the optimal least squares parameter value W. Hint: define matrix A be a diagonal matrix of the weights Q1, ...Qn as follows: [ai 0 ... 0 02 ... 0 | 0 0 ... an] Hint: With this matrix defined, the...
(17 points) Occasionally, a mean function in which the intercept is known to be 0 may be fit. Equivalently, our simple linear regression model is Y; = B1X: + € Assume in a fixed model (z values are fixed), this mean function is given by E[Y] =B120. (a) Assuming € are i.i.d following N(0,02), the residual sum of squares for this model, is SSE = +1(Y; – B12:) Show B1, the least squares estimate of B1. Show that ſi is...
Problem 4: Complete the ANOVA table based on the following data: Replications Standard Deviation Factor A Level 1 Level 2 Mean 15 ANOVA Table Degree Freedom of Sum of square Mean sum of F-valuc Source error square error Treatment Error Total NA NA NA Recall that for single-factor ANOVA, we have the following -SSTreatments+SSp where ss,-Σ Σ(y)-T)-total sum of squares n Σ(vi-r-treatment sum of squares i- SSE _ Σ Σ(w-5)2-error sum of squares Given a dataset like the following: Treatment...
All parts please.
You have the results of a simple linear regression based on state-level data and the (a) The estimated error variance .04672. What is the sum of the squared least (b) The estimated variance of ba is 0.00098. What is the standard error of b2? What 2.7 District of Columbia, a total of N- 51 observations. squares residuals? is the value of Σ(x- 2? (c) Suppose the dependent variable y the state's mean income (in thousands of dollars)...
a) In lecture we derived the estimate of B in WLS as Derive A.wls when p-1. (It should have a form similar to simple linear regression.) (Hints: Notice that we can write a weighted average as analogues of the sums of squares identities we've used; you should derive these if you need to use them.) . You may need to use weighted b) Assume we have the following data: T1 T2 y2 That is, we have a total of n...
Problem 3 The USDA collects data on commodity prices and consumption. Use their data to explain per-capita chicken consumption as a function of the price of chicken. (Note that economic theory suggests that an increase in a commodity price is generally associated with a decrease in demand.) 0per capita chicken consumption PCHICK consumer price index for chicken Collapse Al We estimated a linear regression model using annual data for 1950-2001 (N 52) Dependent variable: coeffidient std. error const PCHICK -0.1...
Problem 3 The USDA collects data on commodity prices and consumption. Use their data to explain per-capita chicken consumption as a function of the price of chicken. (Note that economic theory suggests that an increase in a commodity price is generally associated with a decrease in demand.) Q capita chicken consumption PCHICKconsumer price index for chicken We estimated a linear regression model usng annual data for 1950-2001 ( N = 52 ). Dependent variable: coefficient d. error const 0.05 R-squared:...
Problem 3 The USDA collects data on commodity prices and consumption. Use their data to explain per-capita chicken consumption as a function of the price of chicken. (Note that economic theory suggests that an increase in a commodity price is generally associated with a decrease in demand.) 0 per capita chicken consumption PCHICK consumer price index for chicken We estimated a linear regression model using annual data for 1950-2001 (N 52). Dependent variable: coefficlent std. error const РСНСК -0.1 0.15...
QUESTION 2 Let Xi.. Xn be a random sample from a N (μ, σ 2) distribution, and let S2 and Š-n--S2 be two estimators of σ2. Given: E (S2) σ 2 and V (S2) - ya-X)2 n-l -σ (a) Determine: E S2): (l) V (S2); and (il) MSE (S) (b) Which of s2 and S2 has a larger mean square error? (c) Suppose thatnis an estimator of e based on a random sample of size n. Another equivalent definition of...
The USDA collects data on commodity prices and consumption. Use their data to explain per-capita chicken consumption as a function of the price of chicken. (Note that economic theory suggests that an increase in a commodity price is generally associated with a decrease in demand.) -per capita chicken consumption PCHICK consumer price index for chicken We estimated a linear regression model using annual data for 1950-2001 (N 52) Dependent variable: coefficient std. error const PCHICK -0.1 0.1 R-squared: 0.854880 a....
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