468 economics assignment now add one more observation x101

Question

Economics Assignment -

1. Generate a figure like Figure 2.15 in the textbook for yourself. Begin by drawing 100 observations of a regressor x_t from the N(0, 1) distribution. Then compute and save the h_t for a regression of any regress and on a constant and x_t. Plot the points (x_t, h_t), and you should obtain a graph similar to the one in Figure 2.15.

Now add one more observation, x₁₀₁. Start with x₁₀₁ = x¯, the average value of the x_t, and then increase x₁₀₁ progressively until x₁₀₁ = x¯ + 20. For each value of x₁₀₁, compute the leverage measure h₁₀₁. How does h₁₀₁ change as x₁₀₁ gets larger? Why is this in accord with the result that h_t = 1 if the regressors include the dummy variable e_t?

2. Consider once more the data you found for the last assignment. The model you were asked to estimate is clearly grossly misspecified. See if you can come up with a better model for the consumption function, trying specifications both in levels and in logs, and trying various dynamic models, with lags of both dependent and independent variables.

3. Connect to the site of the St Louis Fed: https://fred.stlouisfed.org, and seek the following series for the US: the interest rate on 90-day treasury bills, (the identifier is TB3MS) p_t, the price index for personal consumption expenditures (PCECTPI), and dy_t, the quarterly percentage change in seasonally adjusted real GDP at annual rates (A191RL1Q225SBEA). The last two series begin in 1947, but for one the first quarter of the year is missing. All have data up to the present, by which I mean the second quarter of 2016. From the price index series, the rate of inflation, πt, can be calculated. Since the interest rate data are monthly, while the other series are quarterly, form a quarterly series, r_t, by averaging the rates over the three months of each quarter. For the period 1948:1 to 2015:4, run the regression

?r_t = β₁ + β₂dy_t + β₃dy_t-1 + β₄π_t + β₅r_t-1 + u_t,

where ? is the first-difference operator, defined so that ?x_t = x_t - x_t-1. Plot the residuals and fitted values against time. Then regress the residuals on the fitted values and on a constant. What do you learn from this second regression? Now regress the fitted values on the residuals and on a constant. What do you learn from this third regression?

Connect to the site of the U.S. Bureau of Economic Analysis, https://www.bea.gov/itable, and seek the series A067RC1 and A071RC1.