Consider the following regression model: Xi = Bo + Bixi + y; where yi is individual i's University GPA and xi is the individual's high school grades. a. What do you think is in ui? Do you think E[ulx) = 0? Suggest a variable that you think might affect University GPA that isn't included in the regression equation but should be. c. What sign of bias would you expect in an OLS regression of y on x? Briefly explain. d....
Consider the model, Yi = Bo + B1 Xi + Uj, where you suspect Xi is endogenous. You have an exogenous instrument and you estimate the first stage to recover the residuals, Vhati. You want to test for endogeneity so you estimate the following model using OLS: Y; = Bo + B1 Xì + B2 Vhat; + Uj. The estimation results from 100 observations are in the table: Coefficient Standard Errors constant 2.96 0.47 X 0.75 0.85 Vhat 0.37 0.15...
Consider the model, Y; = Bo + B1 Xi+Uj, where you suspect Xi is endogenous. You have an exogenous instrument and you estimate the first stage to recover the residuals, Vhatj. You want to test for endogeneity so you estimate the following model using OLS: Y= Bo + B1 Xi + B2 Vhat; + Uj. The estimation results from 100 observations are in the table: Coefficient Standard Errors Constant 2.63 0.98 X 0.97 0.57 Vhat 0.47 0.10 Please select your...
2. Assume the structural equation is where E [ui|Xi] = 0. It was discovered that we observe ri with a measurement error wi instead of the real value X X-Xi + w It is known that E [wi-0, V (wi) %-cou (Xi, wi)-cov is based on regressing Y, on a constant and X. (u,,wi) 0. The OLS estimator (i) Find the value to which the OLS estimator of β¡ is consistent for. (ii) Is the value equal to the true...
y = β0 + β1x + u If yi is earnings of individual i and xi indicates whether individual i is a recent immigrant, what do you think is in u? Do you think E[u|x] = 0? What sign of bias would you expect in an OLS regression of y on x?
4. We have n statistical units. For unit i, we have (xi; yi), for i-1,2,... ,n. We used the least squares line to obtain the estimated regression line у = bo +biz. (a) Show that the centroid (x, y) is a point on the least squares line, where x = (1/n) and у = (1/n) Σ¡ı yi. (Hint: E ) i-1 valuate the line at x = x. (b) In the suggested exercises, we showed that e,-0 and e-0, where...
Consider the regression equation Y = Bo+B1Xi+u; where E[u;|Xi]=0 for all i = 1, ..., n. Let B 1 be the OLS estimator for B 1. Which statement is the most irrelevant to the consistency of B1? Hint: see Lecture Note 2 (p.25-p.28) a. When n is large, the estimator B 1 is near the population parameter B1 O". Consistency of B1 is mathematically written as B1-B1 VB) is inversely proportional to the sample size n. Od. RMSE is close...
R STUDIO Create a simulated bivariate data set consisting of n 100 (xi, yi) pairs: Generate n random a-coordinates c from N(0, 1) Generate n random errors, e, from N(0, o), using o 4. Set yiBoB1x; + , Where Bo = 2, B1 = 3, and eN(0, 4). (That is, y is a linear function of , plus some random noise.) (Now we have simulated data. We'll pretend that we don't know the true y-intercept Bo 2, the true slope...
In the simple linear regression with zero-constant item for (xi , yi) where i = 1, 2, · · · , n, Yi = βxi + i where {i} n i=1 are i.i.d. N(0, σ2 ). (a) Derive the normal equation that the LS estimator, βˆ, satisfies. (b) Show that the LS estimator of β is given by βˆ = Pn i=1 P xiYi n i=1 x 2 i . (c) Show that E(βˆ) = β, V ar(βˆ) = σ...
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 =...