Numerade

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a) Orthogonal Partitioned Regression Theorem In the multiple linear least squares regression of y on two sets of variables X₁ and X₂, if the two sets of variables are orthogonal, then the separate coefficient vectors can be obtained by separate regressions of Y on X₁ alone and y on X₂ alone. Using the necessary notations provide a convincing proof of this theorem. In your proof, you're required to explain the concept of orthogonality in relation to the two sets of variables. b) Frisch-Waugh (1933)-Lovell (1963) Theorem In the linear least squares regression of vector Y on two sets of variables, X₂ and X₁, the subvector $\hat{\beta}_2$ is the set of coefficients obtained when the residuals from a regression of Y on X₁ alone are regressed on the set of residuals obtained when each column of X₂ is regressed on X₁. Using the necessary notations provide a convincing proof of this theorem c) Assume a model (DSP) of the structure $Y = X_1\beta_1 + X_2\beta_2 + u_i$ where $\hat{\beta} = (\hat{\beta}_1', \hat{\beta}_2') = (X'X)^{-1}X'Y$ Here, Y is the dependent variable, X₁ and X₂ are the matrices of the first and second set of regressors respectively, $\beta_1$ and $\beta_2$ are the coefficients to be estimated and $u_i$ is the error term. 1. Show that $\hat{\beta}_1 = (X_1'M_2X_1)^{-1}(X_1'M_2Y)$, where $M_2 = I_n - X_2(X_2'X_2)^{-1}X_2'$. 2. Explain the properties of the matrix $M_2$ and explain how this matrix compares to another similar matrix M. What does the matrix accomplish. 3. How does the matrix M come about? Derive this matrix (residual maker). In the process also derive the projection matrix and explain what the projection matrix accomplishes. d) Suppose a correctly specified model is given by: $Y = X_1\beta_1 + X_2\beta_2 + u_i$ If we by mistake of omission of the key variable we regress Y on X₁ without X₂ so that the misspecified model is: $Y = X_1\beta_1 + v_i$ Show how the bias comes up in the estimation of $\beta_1$ and indicate the actual bias. Explain the special condition or circumstance in which the estimator for $\beta_1$ would be unbiased despite the omission of the key variable, X₂

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7.3. Tara purchases a 20-year, 5 percent savings bond with a face value of $10,000 at a premium price of $11,000. She gets the dividend paid yearly. Determine the rate of return on the bond if the bond is kept till maturity. A) 3.79% B) 4.62% C) 5.33% D) 2.38% 7.4. Pam promises to pay Sandy $4,000 in year 4 and another $6,000 in year 8 for a loan of $4,000 from Sandy today. What is the ROR that Sandy is getting? Assume interest is compounded monthly. A) 14.75% B) 16.72% C) 15.39% D) 18.08% 7.5. An equipment costing $60,000 is being evaluated for a production process at Don Jones Co. The expected benefits per year is $4,500 and estimated salvage value is $20,000. Determine the rate of return the company can get in this equipment proposal. Equipment life = 20 years. A) 4.92% B) 5.96% Year 0 1-2 3-5 Cash flow -$12,300 4,000 3,000 C) 5.62% D) 4.18%

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One important way in which the Federal Reserve Banks provide assistance in the conduct of monetary policy is by Multiple choice question. setting the federal funds rate for their individual districts. buying and selling US treasury securities from the FOMC. providing five Federal Reserve Bank presidents to serve on the FOMC board members. buying or selling US treasury securities from local banks in their individual districts.

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One positive number is 1/5 of another number. The difference between the two numbers is 44. Find the numbers.

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If consumer confidence rises because of strong job growth in an economy, what is likely to happen to the rate of savings? The savings rate will likely fall to zero. The savings rate will likely rise. The savings rate will turn negative. The savings rate will likely fall.

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Experimental Condition Systolic Pressure Diastolic Pressure Pulse Pressure Mean Arterial Pressure Sitting quietly 120 60 120-60 = 60 7801 Laying down 2 minutes 133 85 133-85 = 48 101 Immediately standing 125 85 125-85 =40 98 Standing for 2 minutes 121 65 121-65 =56 84 Questions: 10) Explain the data you recorded immediately on standing after laying down for 2 minutes. 11) How did the MAP change between laying down, immediately on standing and then standing for 2 minutes?

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7. Set up, but do not evaluate, an iterated double integral which equals the volume of the solid bounded by the surfaces $z = 6 - x^2 - y^2$ and $z = 0$ over the rectangle $[-1, 1] \times [0, 2]$ in the $xy$ plane. (7 points)

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An M mass bar and L length is supported by two vertical springs with $k_1$ y $k_2$ constants, separated by a distance of d. If the bar is rotated around an axis passing through its center part, a) What will be the oscillation frequency for damped oscillations, when a b-value damper is placed at the right end of the bar and parallel to the springs? b) Determine the b-value for critically damped oscillations.

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1. For each of the following normal distributions find the standard error of the mean (a) sample size is 36 and standard deviation is 3 (b) sample size is 100 and standard deviation is 2 (c) $\sigma$=4 and n=64 (d) $\sigma$=1 and n=50 2. A sample of n=16 is drawn from a normal distribution with $\mu$=1000 and $\sigma$=200. Find the: (a) $Pr(\bar{x} > 1050)$ (b) $Pr(\bar{x} < 960)$ (c) $Pr(\bar{x} > 1100)$

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72. Zeeman effect. In the Bohr model of the hydrogen atom, the electron is held in its circular orbit of radius $r$ about its proton nucleus by electrostatic attraction. If the atoms are placed in a weak magnetic field $\vec{B}$, the rotation frequency of electrons rotating in a plane perpendicular to $\vec{B}$ is changed by an amount $\Delta f = \pm \frac{eB}{4\pi m}$ where $e$ and $m$ are the charge and mass of an electron. (a) Derive this result, assuming the force due to $\vec{B}$ is much less than that due to electrostatic attraction of the nucleus. (b) What does the $\pm$ sign indicate?

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jennifer w.