Refer to Section   8.8   and the data in Table   8.9   concerning disposable personal income and personal savings for the period 1970-1995. In that section, the Chow test was introduced to see if a structural change occurred within the data between two time periods. Table   8.11   includes updated data containing the values from 1970-2005. According to the National Bureau of Economic Research, the most recent U.S. business contraction cycle ended in late 2001. Split the data into three sections: (1) 1970-1981, (2) 1982–2001, and (3) 2002-2005.
a. Estimate both the model for the full dataset (years 1970-2005) and the third section (post-2002). Using the Chow test, determine if there is a significant break between the third period and the full dataset.
b. With this new data in Table 8.11, determine if there is still a significant difference between the first set of years (1970-1981) and the full dataset, now that there are more observations available.
c. Perform the Chow test on the middle period (1982-2001) versus the full dataset to see if the data in this period behave significantly differently than the rest of the data.>\begin{tabular}{lcccccc} 
TABLE 8.9 & Observation & Savings & Income & Observation & Savings & Income \\
Savings and Personal & 1970 &   61.0   &   727.1   & 1983 &   167.0   &   2522.4   \\
Disposable Income & 1971 &   68.6   &   790.2   & 1984 &   235.7   &   2810.0   \\
(billions of dollars), & 1972 &   63.6   &   855.3   & 1985 &   206.2   &   3002.0   \\
United States, & 1973 &   89.6   &   965.0   & 1986 &   196.5   &   3187.6   \\
1970-1995 & 1974 &   97.6   &   1054.2   & 1987 &   168.4   &   3363.1   \\
& 1975 &   104.4   &   1159.2   & 1988 &   189.1   &   3640.8   \\
Source: Economic Report & 1976 &   96.4   &   1273.0   & 1989 &   187.8   &   3894.5   \\
of the President 1997, & 1977 &   92.5   &   1401.4   & 1990 &   208.7   &   4166.8   \\
Eble B-28,p. 332. & 1978 &   112.6   &   1580.1   & 1991 &   246.4   &   4343.7   \\
& 1979 &   130.1   &   1769.5   & 1992 &   272.6   &   4613.7   \\
& 1980 &   161.8   &   1973.3   & 1993 &   214.4   &   4790.2   \\
& 1981 &   199.1   &   2200.2   & 1994 &   189.4   &   5021.7   \\
& 1982 &   205.5   &   2347.3   & 1995 &   249.3   &   5320.8   \\
\hline
\end{tabular}>8.8 Prediction with Multiple Regression
In Section   5.10   we showed how the estimated two-variable regression model can be used for (1) mean prediction, that is, predicting the point on the population regression function (PRF), as well as for (2) individual prediction, that is, predicting an individual value of   Y   given the value of the regressor   X=$X_{0}$  , where   $X_{0}$   is the specified numerical value of   X  .
The estimated multiple regression too can be used for similar purposes, and the procedure for doing that is a straightforward extension of the two-variable case, except the formulas for estimating the variances and standard errors of the forecast value (comparable to Eqs.   $[5.10 .2]$   and   $[5.10 .6]$   of the two-variable model) are rather involved and are better handled by the matrix methods discussed in Appendix C. Of course, most standard regression packages can do this routinely, so there is no need to look up the matrix formulation. It is given in Appendix   $\mathbf{C}$   for the benefit of the mathematically inclined students. This appendix also gives a fully worked out example.

Refer to Section 8.8 and the data in Table 8.9 concerning disposable personal income and personal savings for the period 1970-1995. In that section, the Chow test was introduced to see if a structural change occurred within the data between two time periods. Table 8.11 includes updated data containing the values from 1970-2005. According to the National Bureau of Economic Research, the most recent U.S. business contraction cycle ended in late 2001. Split the data into three sections: (1) 1970-1981, (2) 1982-2001, and (3) 2002-2005. a. Estimate both the model for the full dataset (years 1970-2005) and the third section (post-2002). Using the Chow test, determine if there is a significant break between the third period and the full dataset. b. With this new data in Table 8.11 , determine if there is still a significant difference between the first set of years (1970-1981) and the full dataset, now that there are more observations available. c. Perform the Chow test on the middle period (1982-2001) versus the full dataset to see if the data in this period behave significantly differently than the rest of the data.
\begin{tabular}{|c|c|c|c|c|c|c|} \hline $\begin{array}{l}\text { TABLE } 8.9 \\ \text { Savings and Personal }\end{array}$ & Observation & Savings & Income & Observation & Savings & Income \\ \hline Disposable Income & 1970 & 61.0 & 727.1 & 1983 & 167.0 & 2522.4 \\ \hline (billions of dollars), & 1971 & 68.6 & 790.2 & 1984 & 235.7 & 2810.0 \\ \hline United States, & 1972 & 63.6 & 855.3 & 1985 & 206.2 & 3002.0 \\ \hline \multirow{2}{*}{$1970-1995$} & 1973 & 89.6 & 965.0 & 1986 & 196.5 & 3187.6 \\ \hline & 1974 & 97.6 & 1054.2 & 1987 & 168.4 & 3363.1 \\ \hline \multirow{8}{*}{$\begin{array}{l}\text { Source: Econamic Report } \\ \text { of the President, 1997, } \\ \text { Eble B.28,p. } 332 .\end{array}$} & 1975 & 104.4 & 1159.2 & 1988 & 189.1 & 3640.8 \\ \hline & 1976 & 96.4 & 1273.0 & 1989 & 187.8 & 3894.5 \\ \hline & 1977 & 92.5 & 1401.4 & 1990 & 208.7 & 4166.8 \\ \hline & 1978 & 112.6 & 1580.1 & 1991 & 246.4 & 4343.7 \\ \hline & 1979 & 130.1 & 1769.5 & 1992 & 272.6 & 4613.7 \\ \hline & 1980 & 161.8 & 1973.3 & 1993 & 214.4 & 4790.2 \\ \hline & 1981 & 199.1 & 2200.2 & 1994 & 189.4 & 5021.7 \\ \hline & 1982 & 205.5 & 2347.3 & 1995 & 249.3 & 5320.8 \\ \hline \end{tabular}
8.8 Prediction with Multiple Regression In Section 5.10 we showed how the estimated two-variable regression model can be used for (1) mean prediction, that is, predicting the point on the population regression function (PRF), as well as for (2) individual prediction, that is, predicting an individual value of $Y$ given the value of the regressor $X=X_{0}$, where $X_{0}$ is the specified numerical value of $X$. The estimated multiple regression too can be used for similar purposes, and the procedure for doing that is a straightforward extension of the two-variable case, except the formulas for estimating the variances and standard errors of the forecast value (comparable to Eqs. [5.10.2] and [5.10.6] of the two-variable model) are rather involved and are better handled by the matrix methods discussed in Appendix C. Of course, most standard regression packages can do this routinely, so there is no need to look up the matrix formulation. It is given in Appendix C for the benefit of the mathematically inclined students. This appendix also gives a fully worked out example.
\begin{tabular}{|c|c|c|c|} \hline $\begin{array}{l}\text { TABLE } 8.11 \\ \text { Savings and Personal }\end{array}$ & Year & Savings & Income \\ \hline Disposable Income & 1970 & 69.5 & 735.7 \\ \hline (billions of dollars), & 1971 & 80.6 & 801.8 \\ \hline United States, & 1972 & 77.2 & 869.1 \\ \hline $1970-2005$ (billions of & 1973 & 102.7 & 978.3 \\ \hline dollars, except as & 1974 & 113.6 & $1,071.6$ \\ \hline noted; quarterly data & 1975 & 125.6 & $1,187.4$ \\ \hline at seasonally adjusted & 1976 & 122.3 & $1,302.5$ \\ \hline annual rates) & 1977 & 125.3 & $1,435.7$ \\ \hline (2) & 1978 & 142.5 & $1,608.3$ \\ \hline Source Deprinent of & 1979 & 159.1 & $1,793.5$ \\ \hline & 1980 & 201.4 & $2,009.0$ \\ \hline & 1981 & 244.3 & $2,246.1$ \\ \hline & 1982 & 270.8 & $2,421.2$ \\ \hline & 1983 & 233.6 & $2,608.4$ \\ \hline & 1984 & 314.8 & $2,912.0$ \\ \hline & 1985 & 280.0 & $3,109.3$ \\ \hline & 1986 & 268.4 & $3,285.1$ \\ \hline & 1987 & 241.4 & $3,458.3$ \\ \hline & 1988 & 272.9 & $3,748.7$ \\ \hline & 1989 & 287.1 & $4,021.7$ \\ \hline & 1990 & 299.4 & $4,285.8$ \\ \hline & 1991 & 324.2 & $4,464.3$ \\ \hline & 1992 & 366.0 & $4,751.4$ \\ \hline & 1993 & 284.0 & $4,911.9$ \\ \hline & 1994 & 249.5 & $5,151.8$ \\ \hline & 1995 & 250.9 & $5,408.2$ \\ \hline & 1996 & 228.4 & $5,688.5$ \\ \hline & 1997 & 218.3 & $5,988.8$ \\ \hline & 1998 & 276.8 & $6,395.9$ \\ \hline & 1999 & 158.6 & $6,695.0$ \\ \hline & 2000 & 168.5 & $7,194.0$ \\ \hline & 2001 & 132.3 & $7,486.8$ \\ \hline & 2002 & 184.7 & $7,830.1$ \\ \hline & 2003 & 174.9 & $8,162.5$ \\ \hline & 2004 & 174.3 & $8,681.6$ \\ \hline & 2005 & 34.8 & $9,036.1$ \\ \hline \end{tabular}