a) for each of the above models, explain why it is not a linear model
b) Prpose a change to model M1 so that it becomes linear
c) Write down the likelihood for n data points (xi1, Yi) from model (M2)
Homework Answers
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a) for each of the above models, explain why it is not a linear
model
b)...
PSTAT 126 meaning a Linear Regression course. 3. Consider three models for a response and two...
PSTAT 126 meaning a Linear Regression course.
3. Consider three models for a response and two predictors xil and xi2, with additive errors єї і.hd. N(0. ơ2). (a) For each of the above models, explain why it is not a linear model in the sense of PSTAT 126. (b) Propose a change to model (M1) so that it becomes a linear model in the sense of PSTAT 126. (c) Write down the likelihood for n data points (xn, Yi) from...
Multivariate linear model Use vector notation to express the Consider the dataset below where a multivariate linear model b and c are three predictors and y is the desired response. 0 9 5 2 What are...
Multivariate linear model Use vector notation to express the Consider the dataset below where a multivariate linear model b and c are three predictors and y is the desired response. 0 9 5 2 What are X and Y in the MMSE exact solution w = (XTX)- XTY? 4 55 10 A multivariate linear model for the response will be expressed as
Multivariate linear model Use vector notation to express the Consider the dataset below where a multivariate linear model...
2. Consider a simple linear regression model for a response variable Yi, a single predictor variable...
2. Consider a simple linear regression model for a response variable Yi, a single predictor variable ri, i-1,... , n, and having Gaussian (i.e. normally distributed) errors Ý,-BzitEj, Ejį.i.d. N(0, σ2) This model is often called "regression through the origin" since E(Yi) 0 if xi 0 (a) Write down the likelihood function for the parameters β and σ2 (b) Find the MLEs for β and σ2, explicitly showing that they are unique maximizers of the likelihood function. (Hint: The function...
3. Determine whether the following are true or false and explain why: a) R2 can be...
3. Determine whether the following are true or false and explain why: a) R2 can be negative. b) R2 can be larger than 1. c) Adjusted R2 can be negative. d) Adjusted R2 can be larger than 1. e) suˆ is a measure of out-of-sample fit f) CV is a measure of out-of-sample fit Now consider the following two models: Yi = β0 + β1Xi + ui (M1) Yi = β0 + β1Xi + β2X2 i + ui (M2) 1...
4. (20 points. You are given the following pairs of observations on (C1, yi), i =...
4. (20 points. You are given the following pairs of observations on (C1, yi), i = 1, 2, 3, 4: 0 Yi 1 1 2 4 3 3 4 5 Consider two linear regression models (Mi) and (M2) given by Y = 1 + [MI] Yi = a + B.Ti + i [Mi] where i=1,2,3, 4. (a) Compute the OLS estimate for call it î) in the linear regression model | M1). (b) Compute the OLS estimates for a and...
2. Consider a simple linear regression i ion model for a response variable Y, a single...
2. Consider a simple linear regression i ion model for a response variable Y, a single predictor variable ,i1.., n, and having Gaussian (i.e. normally distributed) errors: This model is often called "regression through the origin" since E(X) = 0 if xi = 0 (a) Write down the likelihood function for the parameters β and σ2 (b) Find the MLEs for β and σ2, explicitly showing that they are unique maximizers of the likelihood function Hint: The function g(x)log(x) +1-x...
Consider the sample regressions for the linear, the logarithmic, the exponential, and the log-log models. For each of th...
Consider the sample regressions for the linear, the logarithmic,
the exponential, and the log-log models. For each of the estimated
models, predict y when x equals 50. (Do
not round intermediate calculations. Round final answers to 2
decimal places.)
Response Variable: y
Response Variable: ln(y)
Model 1
Model 2
Model 3
Model 4
Intercept
18.52
−6.74
1.48
1.02
x
1.68
NA
0.06
NA
ln(x)
NA
29.96
NA
0.96
se
23.92
19.71
0.12
0.10
Model 1 Model 2 Model 3 Model...
3. Let y = (yi..... Yn) be a set of re- sponses, and consider the linear...
3. Let y = (yi..... Yn) be a set of re- sponses, and consider the linear model y= +E, where u = (1, ..., and e is a vector of zero mean, uncorrelated errors with variance o'. This is a linear model in which the responses have a constant but unknown mean . We will call this model the location model. (a) If we write the location model in the usual form of the linear model y = X 8+...
4) Consider n data points with 2 covariates and observation {xi,i, Vi,2, yi); i -1,... ,n,...
4) Consider n data points with 2 covariates and observation {xi,i, Vi,2, yi); i -1,... ,n, where yi 's are indicator variable for the experiment that is if a particular medicine is effective on some individual. Here, xi1 and ri.2 are age and blood pressure of i th individual, respectively. Our assumption is that the log odds ratio follows a linear model. That is p-P(i-1) and 10i b) What should be a good estimator for ?,A, e) Suppose. On, A,n...
1. Consider the simple linear regression model: Ү, — Во + B а; + Ei, where 1, . . , En are i.i.d. N(0,02), for i1,2,......
1. Consider the simple linear regression model: Ү, — Во + B а; + Ei, where 1, . . , En are i.i.d. N(0,02), for i1,2,... ,n. Let b1 = s^y/8r and bo = Y - b1 t be the least squared estimators of B1 and Bo, respectively. We showed in class, that N(B; 02/) Y~N(BoB1 T;o2/n) and bi ~ are uncorrelated, i.e. o{Y;b} We also showed in class that bi and Y 0. = (a) Show that bo is...
PSTAT 126 meaning a Linear Regression course.
3. Consider three models for a response and two predictors xil and xi2, with additive errors єї і.hd. N(0. ơ2). (a) For each of the above models, explain why it is not a linear model in the sense of PSTAT 126. (b) Propose a change to model (M1) so that it becomes a linear model in the sense of PSTAT 126. (c) Write down the likelihood for n data points (xn, Yi) from...
Multivariate linear model Use vector notation to express the Consider the dataset below where a multivariate linear model b and c are three predictors and y is the desired response. 0 9 5 2 What are X and Y in the MMSE exact solution w = (XTX)- XTY? 4 55 10 A multivariate linear model for the response will be expressed as
Multivariate linear model Use vector notation to express the Consider the dataset below where a multivariate linear model...
2. Consider a simple linear regression model for a response variable Yi, a single predictor variable ri, i-1,... , n, and having Gaussian (i.e. normally distributed) errors Ý,-BzitEj, Ejį.i.d. N(0, σ2) This model is often called "regression through the origin" since E(Yi) 0 if xi 0 (a) Write down the likelihood function for the parameters β and σ2 (b) Find the MLEs for β and σ2, explicitly showing that they are unique maximizers of the likelihood function. (Hint: The function...
4. (20 points. You are given the following pairs of observations on (C1, yi), i = 1, 2, 3, 4: 0 Yi 1 1 2 4 3 3 4 5 Consider two linear regression models (Mi) and (M2) given by Y = 1 + [MI] Yi = a + B.Ti + i [Mi] where i=1,2,3, 4. (a) Compute the OLS estimate for call it î) in the linear regression model | M1). (b) Compute the OLS estimates for a and...
2. Consider a simple linear regression i ion model for a response variable Y, a single predictor variable ,i1.., n, and having Gaussian (i.e. normally distributed) errors: This model is often called "regression through the origin" since E(X) = 0 if xi = 0 (a) Write down the likelihood function for the parameters β and σ2 (b) Find the MLEs for β and σ2, explicitly showing that they are unique maximizers of the likelihood function Hint: The function g(x)log(x) +1-x...
Consider the sample regressions for the linear, the logarithmic,
the exponential, and the log-log models. For each of the estimated
models, predict y when x equals 50. (Do
not round intermediate calculations. Round final answers to 2
decimal places.)
Response Variable: y
Response Variable: ln(y)
Model 1
Model 2
Model 3
Model 4
Intercept
18.52
−6.74
1.48
1.02
x
1.68
NA
0.06
NA
ln(x)
NA
29.96
NA
0.96
se
23.92
19.71
0.12
0.10
Model 1 Model 2 Model 3 Model...
3. Let y = (yi..... Yn) be a set of re- sponses, and consider the linear model y= +E, where u = (1, ..., and e is a vector of zero mean, uncorrelated errors with variance o'. This is a linear model in which the responses have a constant but unknown mean . We will call this model the location model. (a) If we write the location model in the usual form of the linear model y = X 8+...
4) Consider n data points with 2 covariates and observation {xi,i, Vi,2, yi); i -1,... ,n, where yi 's are indicator variable for the experiment that is if a particular medicine is effective on some individual. Here, xi1 and ri.2 are age and blood pressure of i th individual, respectively. Our assumption is that the log odds ratio follows a linear model. That is p-P(i-1) and 10i b) What should be a good estimator for ?,A, e) Suppose. On, A,n...
1. Consider the simple linear regression model: Ү, — Во + B а; + Ei, where 1, . . , En are i.i.d. N(0,02), for i1,2,... ,n. Let b1 = s^y/8r and bo = Y - b1 t be the least squared estimators of B1 and Bo, respectively. We showed in class, that N(B; 02/) Y~N(BoB1 T;o2/n) and bi ~ are uncorrelated, i.e. o{Y;b} We also showed in class that bi and Y 0. = (a) Show that bo is...
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