Simple linear regression is a statistical method that allows us to summarize and study relationships between two continuous (quantitative) variables: One variable, denoted X, is regarded as the predictor, explanatory, or independent variable. The other variable, denoted Y, is regarded as the response, outcome, or dependent variable. Suppose that we are given n-i.i.d observations { (x;, y;)}"_, from the assumed simple linear regression model Y = BIX + Bo + . Answer the following questions on simple linear regression. 5-a. Denote 1 and Bo as the point estimators of B, and Bo, respectively, that are obtained through the least squares method. Show, step by step, that the two point estimators are unbiased. Derive the least squares estimator of of and determine whether it is unbiased or not. Show your work step by step. 5-b. Calculate _'_1(yi - Bix; - Bo) (Bix, + Bo). Determine whether the point (X, Y) is on the line Y = 1X + Bo. Explain your reasoning mathematically. 5-c. Using the maximum likelihood estimation (MLE) technique, derive a point estimator for the coefficient B1 and the intercept Bo, respectively. Determine whether the point estimators that you obtained via MLE are unbiased or not. Justify your conclusion mathematically. 5-d. Calculate the variance of the four estimators from Questions 5-a and 5-c, respectively. Show your work step by step. 5-e. Suppose that we are using the simple linear regression model Y = B1 X + Bo + 1 while the true model is Y = 1X1 + B2X2 + Bo + 82 where Bo, B1, and B2 are constants. We assume that the distributions of &, and e2 are both N(0,02), i.e., normal distribution with variance o?. We further assume that the two noise variables are uncorrelated. Find the least squares estimator of B, in this case and determine whether the point estimator that you obtain is biased or not. If it is biased, calculate the bias.2. At this point, we can analyze (stability, steady-state gain, sinusoidal steady-state gains, time-constant, etc.) of first-order, linear dynamical systems. We previously analyzed a Ist-order process model, and a proportional-control strategy. In this problem, we try a different situation, where the process is simply proportional, but the controller is a Ist-order, linear dynamical system. Specifically, suppose the process model is non-dynamic ("static" ) simply y(t) = cu(t) + Bd(t) where o and B are constants. The control strategy is dynamic i (t) = ar(t) + bir(t) + bzym(t) u(t) = cr(t) + dir(t) where ym(t) = y(t) + n(t) and the various "gains" (a, bi, . .., di) constitute the design choices in the control strategy. Be careful, notation-wise, since (for example) d, is a constant parameter, and d(t) is a signal (the disturbance). (a) Eliminate u and ym from the equations to obtain a differential equation for r of the form r(t) = Ar(t) + Bir(t) + Bad(t) + Ban(t) which governs the closed-loop behavior of r. Note that A, B1, B2, By are functions of the parameters a, b1, ... in the control strategy, as well as the process parameters o and B. (b) What relations on (a, b1. .... dj, or, B) are equivalent to closed-loop system stability? (c) As usual, we are interested in the effect (with feedback in place) of (r, d, n) on (y, u), the regulated variable, and the control variable, respectively. Find the coefficients (in terms of (a, bi, . . ., d1, 0, B)) so that y(t) = Cix(t) + Dur(t) + Died(t) + Dian(t) u(t) = Car(t) + Dar(t) + Dad(t) + Dzan(t) (d) Suppose that T. > 0 is a desired closed-loop time constant. Write down the constraints on the a, b1, b2, c and di (i.e., the parameters of the controller to be design) such that the following conditions hold: . closed-loop is stable . closed-loop time constant is To . steady-state gain from d -> y is 0 . steady-state gain from r - y is 12. At this point, we can analyze (stability, steady-state gain, sinusoidal steady-state gains, time-constant, etc.) of first-order, linear dynamical systems. We previously analyzed a Ist-order process model, and a proportional-control strategy. In this problem, we try a different situation, where the process is simply proportional, but the controller is a Ist-order, linear dynamical system. Specifically, suppose the process model is non-dynamic ("static" ) simply y(t) = cu(t) + Bd(t) where o and B are constants. The control strategy is dynamic i (t) = ar(t) + bir(t) + bzym(t) u(t) = cr(t) + dir(t) where ym(t) = y(t) + n(t) and the various "gains" (a, bi, . .., di) constitute the design choices in the control strategy. Be careful, notation-wise, since (for example) d, is a constant parameter, and d(t) is a signal (the disturbance). (a) Eliminate u and ym from the equations to obtain a differential equation for r of the form r(t) = Ar(t) + Bir(t) + Bad(t) + Ban(t) which governs the closed-loop behavior of r. Note that A, B1, B2, By are functions of the parameters a, b1, ... in the control strategy, as well as the process parameters o and B. (b) What relations on (a, b1. .... dj, or, B) are equivalent to closed-loop system stability? (c) As usual, we are interested in the effect (with feedback in place) of (r, d, n) on (y, u), the regulated variable, and the control variable, respectively. Find the coefficients (in terms of (a, bi, . . ., d1, 0, B)) so that y(t) = Cix(t) + Dur(t) + Died(t) + Dian(t) u(t) = Car(t) + Dar(t) + Dad(t) + Dzan(t) (d) Suppose that T. > 0 is a desired closed-loop time constant. Write down the constraints on the a, b1, b2, c and di (i.e., the parameters of the controller to be design) such that the following conditions hold: . closed-loop is stable . closed-loop time constant is To . steady-state gain from d -> y is 0 . steady-state gain from r - y is 1vi payments accounts, still PLEsuit IsOnly foreign currency traded with the Canadian dollar. 7. For each of the following situations, outline the effect on the price of the Canadian dollar in terms of US dollars and draw a demand and supply graph that illustrates the changes that occur in the foreign exchange market for the Canadian dollar. . A contractionary monetary policy initiated by the Bank of Canada raises Canadian interest rates. b. Canada's real output rises at a time when real output in the United States is falling. c. Americans (but not Canadians) find Canada a more attractive place to make financial investments. d. Given Canada's aging population, more Canadian "snowbirds" travel to the United States each winter. e. Due to a credit crisis that affects US financial institutions more than it does Canadian ones, Canada's attractiveness as a destination for direct and portfolio investment increases. f. The Bank of Canada initiates an expansionary monetary policy that reduces Canadian interest rates