(Econometrics) 1. Consider the simple linear model and let Z, be a binary instrumental variable for...
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...
(Econometrics | Instrumental Variables) Consider the model: Some researchers have suggested using an instrumental variables (IV) approach to estimating ß1. Specifically, a recent study suggested using the variable Catholic, which is equal to 1 if a student is Catholic and zero if not, as an instrumental variable for privateHS. What two conditions does Catholic have to satisfy in order to deliver consistent estimates of ß1? Which of these can be formally tested, and how? Be specific. college-BiprivateHS+u college-BiprivateHS+u
Let X, denote a binary variable and consider the regressions Yi = A + Ax, + ui , Let Yo denote the sample mean for observations with X0 and let Y1 denote the sample mean for observations with X-1. Show that β,-Ý, Άο + β,-Ý, , and A-R-Ý, 6.
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...
Question 1 1 pts Consider the simple, general model: Y = Bo + BX; + i. A proposed instrumental variables, Zi, is relevant if Z and X are uncorrelated Z and u are correlated Z and u are uncorrelated Zand X are correlated
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 simple linear regression model: Yi = Bo + Bilitei, i = 1,...,n. with the least squares estimates ỘT = (Bo ß1). We observe a new value of the predictor: x] = (1 xo). Show that the expression for the 100(1 - a)% prediction interval reduces to the following: . (xo – x2 Ēo + @130 Etap 11+ntan (x; – 7)2
1. If a true model of simple linear regression reads: yi −y ̄ = β0 +β1(xi −x ̄)+εi for i = 1, 2, · · · , n, showβ0 =0andβˆ0 =0. (1pt) (hint: use the formula of estimator βˆ0 = y ̄ − βˆ1x ̄.)
Consider a simple linear regression model with nonstochastic regressor: Yi = β1 + β2Xi + ui. 1. [3 points] What are the assumptions of this model so that the OLS estimators are BLUE (best linear unbiased estimates)? 2. [4 points] Let βˆ and βˆ be the OLS estimators of β and β . Derive βˆ and βˆ. 12 1212 3. [2 points] Show that βˆ is an unbiased estimator of β .22
Consider a simple model to measure the effects of taking a preparatory course (a binary variable, course) on eventual score on a college admissions exam: score = Bo + Bicourse + u a. If we want to test whether or not the preparatory course has an effect on the score of the college admissions exam, what are our null and alternative hypotheses? When do we reject the null hypothesis? b. Why might course be correlated with u? Is ê, likely...