Suppose that the error term u in regression y = Bo + B1x + u is in- dependent of the explanatory variable x, and it takes on the values -2, –1,0, 1, and 2 with equal probability of 1/5. Select TRUE or FALSE: (a) The Gauss-Markov assumptions are violated. (b) Normal distribution assumption of the error term is violated.
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- Consider a simple linear regression model, Y₁ = Bo + B₁X₁ + ₁ for i=1,2,...,n with the usual random error term (&) assumptions. Suppose that hypotheses Ho: P₁ = 0 against H₁:₁ <0 were tested and Ho: B₁ = 0 was rejected. Assume ₁e₁² = 0. ei i) Briefly explain what the conclusion of the test means for the values of Y given the values of X. -1 ii) Draw an example of a scatter plot with the fitted regression line for the problem mentioned above. iii) Calculate and interpret the correlation coefficient for the problem at hand.Consider the following population model that satisfies the CNLRM assumptions: Y = βX; + u in particular assume that: ui ~ Ν (0,1) What is the mean and variance of B₁? Ε (βι) = 0,6%, = nΣα 22 x Στ Ο Ο Ο ΣΧ Ε (βι) = 0,6% = nΣ # Ε(β) = 1,6 = Ε(βι) = 1,6% = nΣ # (31) {" 2 Σ βι η ΣΧ Cannot be determined with the information provided.1.
- Consider the multiple regression model Y a + B1 X1 + ß2 X2 + u . When omitting X2 from the regression, then there will be omitted variable bias for B Only if X1 and X2 are correlated, and ß1 # 0 Only if X1 and X2 are correlated, and B2 + 0 Only if X1 and X2 are correlated, ß1 # 0 and B2 # 0 O Only if ß1 + 0 and ß2 # 0For ALL the following statements, evaluate each statement as either TRUE or FALSE. Then, justify your answer with a careful explanation. Please note that explanations may also involve mathematical and/or graphical illustrations.1 Consider an estimated linear regression model with a response Y and four predictors X1, X2, X3, and X4, based on a random sample of 25 sales agents of medical supplies in 3 regions of the province. Y is annual sales in 1000$, X1 in 100$, X2 in 100$, X3 in %, and X4 in 1000$. The regression results are given below. Coefficients Standard Error Intercept -593.53745 259.19585 2.51314 0.31428 1.90595 0.74239 2.65101 4.63566 -0.12073 0.37181 X1 X2 X3 X4 Type your numerical answer here. Numbers only Find the lower limit of the change in the mean response in dollars, for a unit change in X1 with a 95% confidence level, assuming that the other predictors remain unchanged over the period.
- In this problem, we consider target detection in a radar system. Upon observing a realization a of the random variable X which represents the return signal in noise, a decision has to be made as to whether the target is present or absent. The target amplitude is known and equal to A > 0. The observation is made in zero-mean Gaussian noise W whose variance is known and equal to 2. If the target is present (hypothesis H1), X = A+W. If the target is absent (hypothesis Ho), X = W . The likelihood ratio test for this problem is stated as follows: If fi (x)/fo(x) > n= decide in favor of H1; otherwise, decide in favor of Ho.lf n = 1, determine the probability of correct detection (target declared present when indeed present) and the probability of false alarm (target declared present when in fact absent). Express these probabilities in terms of the ratio a = A/o. If a = 4, which of the following represents the correct probabilities of false alarm and correct detection, respectively? O None of…5Consider the regression model y = a + ßx+u, where x is an endogenous variable. a) Write x₁ = x + ê; and show that = x. b) Suppose there are k valid instruments: Z1, Z2, ..., Zk. Let x₁= πo+f1zil + π2Zi2+...+îkzik. Show that the 2SLS estimator is equivalent to an IV estimator using as an instrument: Σ₁₁(&i- Ñ)(vi- V) _ Σ₁(Ri—Ñ)(Yi-Y) Ei=1(xi-x)² 21(®i- Đ)(xi-x) = [Hint: OLS first order conditions imply №₁ ê¡Â¡ = 0 and Σ¼_₁ ê¡= 0.] c) Suppose there is only one instrument z and ✰; = fo+ fizi. Show that the 2SLS estimator is numerically identical to the IV estimator: = Σ₁ (Îi—Ñ)(vi-V) – Σ₁(²₁-7)(vi-V) Σ=1(ât− x)2 Σ₁ (Z₁-7)(xi-x)* n [Hint: Use the result in (b) and plug in â; and ☎ = ñîo+â¦Z.]
- 1. Derive the least squares estimators (LSES) of the parameters in the simple linear regression model. 2. Derive the estimators of 30 and ß1 using maximum likelihood estimation procedures.Phoebe gathers data and estimates many di¤erent regression models. All of them suggest that children who have more books at home have fewer cavities. Phoebe doesnt think this result is important because: a.) There is clearly selection bias, kids who have lots of books at come from higher income and better educated families.b.) The data on cavities is certainly heteroskedasticc.) Phoebe only used ordinary least squares in her model, should have used weighted least squares d.) Phoebe didnt use a high enough con dence level1. Consider the following regression model y = x3 + u. (1) Let 3 denote the Ordinary Least Squares (OLS) estimator of B. The so-called Gauss- Markov assumptions are: • MLR.1: The true model in the population is given by (1). • MLR.2: We have a random sample of n observations {(ri, Yi), i = 1, 2, .., n} following the population model in (1). .... • MLR.3: No one explanatory variable can be written as a linear combination of the remaining explanatory variables that is, there is no perfectcollinearity. • MLR.4: In the population, the error u has an expected value of zero given any values of the explanatory variables, that is Elu|x] = 0. • MLR.5: In the population, the error u has the same variance given any values of the explanatory variables, that is Var[u|x] = o? , an unknown finite, positive constant. In the following scenarios, state whether 3 is an unbiased and consistent estimator of 3, and provide a brief justification for your answer in each case - but no formal mathematical…
