Increases in the mortgage interest rate increase the cost of owning a house and lower the demand for houses. In this question, we use three equations to forecast the monthly change in the number of new one-family houses sold in the United States. In the first equation (XR 9.12.1), the monthly change in the number of houses DHOMES is regressed against two lags of the monthly change in the 30-year conventional mortgage rate DIRATE. In the second equation (XR 9.12.2), DHOMES is regressed against two lags of itself, and in the third equation (XR 9.12.3), two lags of both DHOMES and DIRATE are included as regressors.

The data used are from January, 1992 (1992M1) to September, 2016 (2016M9). The units of measurement are thousands of new houses for DHOMES and percentage points for DIRATE. After differencing and allowing for two lags, three observations are lost, resulting in a total of 294 observations that were used to produce the least squares estimates in Table 9.11.

a. Given DHOMES D \(_{2016 \mathrm{M} 8}=-54\), DHOMES \(_{2016 \mathrm{M} 9}=18\), DIRATE \(_{2016 \mathrm{M} 8}=0.00\), DIRATE \(_{2016 \mathrm{M} 9}=\) 0.02 , and DIRATE 2016M10 \(=-0.01\), use each of the three estimated equations to find \(95 \%\) forecast intervals for DHOMES \(S_{2016 \mathrm{M} 10}\) and \(D H O M E S_{2016 \mathrm{M} 11}\). Comment on the results.

b. Using a \(5 \%\) significance level, test for autocorrelated errors in each of the equations.

c. Using a 5\% significance level, test whether DIRATE Granger causes DHOMES.

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DHOMES, = 8+8, DIRATE, +8DIRATE,-2 + 1,1 (XR 9.12.1) (XR 9.12.2) (XR 9.12.3) DHOMES, = 8+0,DHOMES,1 +0DHOMES,_2+ 21 DHOMES,=8+8,DHOMES +8,DHOMES-2 +8, DIRATE, +8DIRATE-2 + 3,1