Question #51

The one-period forecast of a random walk model without drift is E(xt+1) = E(xt + et ) = xt + 0, so the forecast is simply xt = 2.2. For a random walk process, the variance changes with the value of the observation. However, the error term et = xt - xt-1 is not autocorrelated. (Module 2.3, LOS 2.i) Housing industry analyst Elaine Smith has been assigned the task of forecasting housing foreclosures. Specifically, Smith is asked to forecast the percentage of outstanding mortgages that will be foreclosed upon in the coming quarter. Smith decides to employ multiple linear regression and time series analysis. Besides constructing a forecast for the foreclosure percentage, Smith wants to address the following two questions: Research Question 1: Is the foreclosure percentage significantly affected by short-term interest rates? Research Question 2: Is the foreclosure percentage significantly affected by government intervention policies? Smith contends that adjustable rate mortgages often are used by higher risk borrowers and that their homes are at higher risk of foreclosure. Therefore, Smith decides to use short- term interest rates as one of the independent variables to test Research Question 1. To measure the effects of government intervention in Research Question 2, Smith uses a dummy variable that equals 1 whenever the Federal government intervened with a fiscal policy stimulus package that exceeded 2% of the annual Gross Domestic Product. Smith sets the dummy variable equal to 1 for four quarters starting with the quarter in which the policy is enacted and extending through the following 3 quarters. Otherwise, the dummy variable equals zero. Smith uses quarterly data over the past 5 years to derive her regression. Smith's regression equation is provided in Exhibit 1: Exhibit 1: Foreclosure Share Regression Equation foreclosure share = b0 + b1(ΔINT) + b2(STIM) + b3(CRISIS) + ε where: Foreclosure share = the percentage of all outstanding mortgages foreclosed upon during the quarter ΔINT = the quarterly change in the 1-year Treasury bill rate (e.g., ΔINT = 2 for a two percentage point increase in interest rates) STIM = 1 for quarters in which a Federal fiscal stimulus package was in place CRISIS = 1 for quarters in which the median house price is one standard deviation below its 5-year moving average The results of Smith's regression are provided in Exhibit 2: Exhibit 2: Foreclosure Share Regression Results Variable Coefficient t-statistic Intercept 3.00 2.40 ΔINT 1.00 2.22 STIM -2.50 -2.10 CRISIS 4.00 2.35 The ANOVA results from Smith's regression are provided in Exhibit 3: Exhibit 3: Foreclosure Share Regression Equation ANOVA Table Source Degrees of Freedom Sum of Squares Mean Sum of Squares Regression 3 15 5.0000 Error 16 5 0.3125 Total 19 20 Smith expresses the following concerns about the test statistics derived in her regression: Concern 1: If my regression errors exhibit conditional heteroskedasticity, my t- statistics will be underestimated. Concern 2: If my independent variables are correlated with each other, my F-statistic will be overestimated. Before completing her analysis, Smith runs a regression of the changes in foreclosure share on its lagged value. The following regression results and autocorrelations were derived using quarterly data over the past 5 years ( Exhibit 4 and Exhibit 5, respectively): Exhibit 4. Lagged Regression Results Δ foreclosure sharet = 0.05 + 0.25(Δ foreclosure sharet– 1) Exhibit 5. Autocorrelation Analysis Lag Autocorrelation t-statistic 1 0.05 0.22 2 -0.35 -1.53 3 0.25 1.09 4 0.10 0.44 Exhibit 6 provides critical values for the Student's t-Distribution Exhibit 6: Critical Values for Student's t-Distribution Area in Both Tails Combined Degrees of Freedom 20% 10% 5% 1% 16 1.337 1.746 2.120 2.921 17 1.333 1.740 2.110 2.898 18 1.330 1.734 2.101 2.878 19 1.328 1.729 2.093 2.861 20 1.325 1.725 2.086 2.845