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? Explain
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Consider the following formulations of the 1 variable regression
model:
Y = β0 + β1x +...
(Do this problem without using R) Consider the simple linear regression model y =β0 + β1x...
(Do this problem without using R) Consider the simple linear regression model y =β0 + β1x + ε, where the errors are independent and normally distributed, with mean zero and constant variance σ2. Suppose we observe 4 observations x = (1, 1, −1, −1) and y = (5, 3, 4, 0). (a) Fit the simple linear regression model to this data and report the fitted regression line. (b) Carry out a test of hypotheses using α = 0.05 to determine...
1. Consider the following simple regression model: y = β0 + β1x1 + u (1) and...
1. Consider the following simple regression model: y = β0 + β1x1 + u (1) and the following multiple regression model: y = β0 + β1x1 + β2x2 + u (2), where x1 is the variable of primary interest to explain y. Which of the following statements is correct? a. When drawing ceteris paribus conclusions about how x1 affects y, with model (1), we must assume that x2, and all other factors contained in u, are uncorrelated with x1. b....
An economist estimates the following model: y = β0 + β1x + ε. She would like...
An economist estimates the following model: y = β0 + β1x + ε. She would like to construct interval estimates for y when x equals 2. She estimates a modified model where y is the response variable and the explanatory variable is now defined as x+ = x – 2. A portion of the regression results is shown in the accompanying table. Regression Statistics R Square 0.42 Standard Error 3.86 Observations 12 Coefficients Standard Error t-stat p-value Lower 95% Upper...
Suppose that you fitted the model E(y) = β0 + β1x + β2x2 to n =...
Suppose that you fitted the model E(y) = β0 + β1x + β2x2 to n = 20 data points and obtained the following MINITAB printout. Regression Analysis: y versus x, x-sq Analysis of Variance Source DF Adj SS Adj MS F-Value P-Value Regression 2 41225.4 20612.7 987.09 0.000 Error 17 355.0 20.9 Total 19 41580.4 Model Summary S R-Sq R-Sq(adj) 4.56972 99.15% 99.05% Coefficients Term Coef SE Coef T-Value P-Value Constant 12.53 3.40 3.69 0.002 x 9.74 1.49 6.54 0.000...
Consider the linear regression model Yi = β0 + β1 Xi + ui Yi is the...
Consider the linear regression model Yi = β0 + β1 Xi + ui Yi is the ______________, the ______________ or simply the ______________. Xi is the ______________, the ______________ or simply the ______________. is the population regression line, or the population regression function. There are two ______________ in the function (β0 & β1 ). β0 is is the ______________ of the population regression line; β1is is the ______________ of the population regression line; and ui is the ______________. A. Coefficients...
Decide (with short explanations) whether the following statements are true or false. e) In a simple linear regression model with explanatory variable x and outcome variable y, we have these summary s...
Decide (with short explanations) whether the following
statements are true or false.
e) In a simple linear regression model with explanatory variable x and outcome variable y, we have these summary statisties z-10, s/-3 sy-5 and у-20. For a new data point with x = 13, it is possible that the predicted value is y = 26. f A standard multiple regression model with continuous predictors and r2, a categorical predictor T with four values, an interaction between a and...
4. Suppose we run a regression model Y = β0+AX+U when the true model is Y-a0+ α1X2 + V. Assume th...
a,b,c,d
4. Suppose we run a regression model Y = β0+AX+U when the true model is Y-a0+ α1X2 + V. Assume that the true model satisfies all five standard assumptions of a simple regression model discussed in class. (a) Does the regression model we are running satisfy the zero conditional mean assumption? (b) Find the expected value of A (given X values). (e) Does the regression model we are running satisfy homoscedasticity? d) Find the variance of pi (given X...
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...
1. Functional form misspecification and RESET Consider the following model that satisfies assumption MLR.4: y=β0+β1x1+. ....
1. Functional form misspecification and RESET Consider the following model that satisfies assumption MLR.4: y=β0+β1x1+. . .+βkxk+u Which of the following describes the regression specification error test (RESET)? PICK all that apply. RESET picks up all kinds of neglected nonlinearities when more quadratic terms are added to the original model. RESET works better when there are many explanatory variables in the original model, as it increases its degrees of freedom. To implement RESET, the researcher must add at least seven...
Decide (with short explanations) whether the following
statements are true or false.
e) In a simple linear regression model with explanatory variable x and outcome variable y, we have these summary statisties z-10, s/-3 sy-5 and у-20. For a new data point with x = 13, it is possible that the predicted value is y = 26. f A standard multiple regression model with continuous predictors and r2, a categorical predictor T with four values, an interaction between a and...
a,b,c,d
4. Suppose we run a regression model Y = β0+AX+U when the true model is Y-a0+ α1X2 + V. Assume that the true model satisfies all five standard assumptions of a simple regression model discussed in class. (a) Does the regression model we are running satisfy the zero conditional mean assumption? (b) Find the expected value of A (given X values). (e) Does the regression model we are running satisfy homoscedasticity? d) Find the variance of pi (given X...
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...
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