In: Statistics and Probability
Consider a portion of monthly return data (In %) on 20-year Treasury Bonds from 2006–2010.
Date | Return |
Jan-06 | 3.13 |
Feb-06 | 4.15 |
⋮ | ⋮ |
Dec-10 | 4.48 |
Source: Federal Reserve Bank of Dallas.
Estimate a linear trend model with seasonal dummy variables to make forecasts for the first three months of 2011. (Round answers to 2 decimal places.)
Year | Month | yˆty^t |
2011 | Jan | |
2011 | Feb | |
2011 | Mar | |
DATA:
Index | Month | Year | Return |
1 | Jan | 2006 | 3.13 |
2 | Feb | 2006 | 4.15 |
3 | Mar | 2006 | 3.18 |
4 | Apr | 2006 | 4.94 |
5 | May | 2006 | 4.34 |
6 | Jun | 2006 | 4.19 |
7 | Jul | 2006 | 5.12 |
8 | Aug | 2006 | 5.26 |
9 | Sep | 2006 | 3.81 |
10 | Oct | 2006 | 3.1 |
11 | Nov | 2006 | 3.87 |
12 | Dec | 2006 | 4.89 |
13 | Jan | 2007 | 3.94 |
14 | Feb | 2007 | 3.42 |
15 | Mar | 2007 | 4.13 |
16 | Apr | 2007 | 3.54 |
17 | May | 2007 | 4.58 |
18 | Jun | 2007 | 4.19 |
19 | Jul | 2007 | 4.62 |
20 | Aug | 2007 | 3.89 |
21 | Sep | 2007 | 3.62 |
22 | Oct | 2007 | 3.92 |
23 | Nov | 2007 | 4.46 |
24 | Dec | 2007 | 3.23 |
25 | Jan | 2008 | 4.78 |
26 | Feb | 2008 | 4.71 |
27 | Mar | 2008 | 5.05 |
28 | Apr | 2008 | 3.46 |
29 | May | 2008 | 3.15 |
30 | Jun | 2008 | 4.82 |
31 | Jul | 2008 | 3.87 |
32 | Aug | 2008 | 3.78 |
33 | Sep | 2008 | 3.22 |
34 | Oct | 2008 | 5.39 |
35 | Nov | 2008 | 4.78 |
36 | Dec | 2008 | 5.5 |
37 | Jan | 2009 | 4.8 |
38 | Feb | 2009 | 5.2 |
39 | Mar | 2009 | 3.82 |
40 | Apr | 2009 | 4.52 |
41 | May | 2009 | 3.53 |
42 | Jun | 2009 | 4.66 |
43 | Jul | 2009 | 5.46 |
44 | Aug | 2009 | 3.49 |
45 | Sep | 2009 | 3.75 |
46 | Oct | 2009 | 4.84 |
47 | Nov | 2009 | 4.83 |
48 | Dec | 2009 | 4.35 |
49 | Jan | 2010 | 4.63 |
50 | Feb | 2010 | 5.32 |
51 | Mar | 2010 | 4.75 |
52 | Apr | 2010 | 3.28 |
53 | May | 2010 | 4.8 |
54 | Jun | 2010 | 3.21 |
55 | Jul | 2010 | 4.4 |
56 | Aug | 2010 | 3.31 |
57 | Sep | 2010 | 4.81 |
58 | Oct | 2010 | 5.4 |
59 | Nov | 2010 | 3.54 |
60 | Dec | 2010 | 4.48 |
for linear trend | seasonal dummies for seasonal variation | ||||||||||||||
Index | Month | Year | Return(yt) | period(t) | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 |
1 | Jan | 2006 | 3.13 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
2 | Feb | 2006 | 4.15 | 2 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
3 | Mar | 2006 | 3.18 | 3 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
4 | Apr | 2006 | 4.94 | 4 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
5 | May | 2006 | 4.34 | 5 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
6 | Jun | 2006 | 4.19 | 6 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
7 | Jul | 2006 | 5.12 | 7 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
8 | Aug | 2006 | 5.26 | 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
9 | Sep | 2006 | 3.81 | 9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
10 | Oct | 2006 | 3.1 | 10 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
11 | Nov | 2006 | 3.87 | 11 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
12 | Dec | 2006 | 4.89 | 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
13 | Jan | 2007 | 3.94 | 13 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
14 | Feb | 2007 | 3.42 | 14 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
15 | Mar | 2007 | 4.13 | 15 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
16 | Apr | 2007 | 3.54 | 16 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
17 | May | 2007 | 4.58 | 17 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
18 | Jun | 2007 | 4.19 | 18 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
19 | Jul | 2007 | 4.62 | 19 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
20 | Aug | 2007 | 3.89 | 20 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
21 | Sep | 2007 | 3.62 | 21 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
22 | Oct | 2007 | 3.92 | 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
23 | Nov | 2007 | 4.46 | 23 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
24 | Dec | 2007 | 3.23 | 24 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
25 | Jan | 2008 | 4.78 | 25 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
26 | Feb | 2008 | 4.71 | 26 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
27 | Mar | 2008 | 5.05 | 27 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
28 | Apr | 2008 | 3.46 | 28 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
29 | May | 2008 | 3.15 | 29 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
30 | Jun | 2008 | 4.82 | 30 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
31 | Jul | 2008 | 3.87 | 31 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
32 | Aug | 2008 | 3.78 | 32 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
33 | Sep | 2008 | 3.22 | 33 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
34 | Oct | 2008 | 5.39 | 34 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
35 | Nov | 2008 | 4.78 | 35 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
36 | Dec | 2008 | 5.5 | 36 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
37 | Jan | 2009 | 4.8 | 37 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
38 | Feb | 2009 | 5.2 | 38 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
39 | Mar | 2009 | 3.82 | 39 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
40 | Apr | 2009 | 4.52 | 40 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
41 | May | 2009 | 3.53 | 41 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
42 | Jun | 2009 | 4.66 | 42 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
43 | Jul | 2009 | 5.46 | 43 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
44 | Aug | 2009 | 3.49 | 44 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
45 | Sep | 2009 | 3.75 | 45 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
46 | Oct | 2009 | 4.84 | 46 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
47 | Nov | 2009 | 4.83 | 47 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
48 | Dec | 2009 | 4.35 | 48 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
49 | Jan | 2010 | 4.63 | 49 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
50 | Feb | 2010 | 5.32 | 50 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
51 | Mar | 2010 | 4.75 | 51 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
52 | Apr | 2010 | 3.28 | 52 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
53 | May | 2010 | 4.8 | 53 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
54 | Jun | 2010 | 3.21 | 54 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
55 | Jul | 2010 | 4.4 | 55 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
56 | Aug | 2010 | 3.31 | 56 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
57 | Sep | 2010 | 4.81 | 57 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
58 | Oct | 2010 | 5.4 | 58 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
59 | Nov | 2010 | 3.54 | 59 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
60 | Dec | 2010 | 4.48 | 60 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
The regression output for above data taking return as dependent variable(y) and taking period(t) and seasonal dummies as independent variables is given below
Coefficients | Standard Error | t Stat | P-value | |
Intercept | 4.24975 | 0.383441215 | 11.08319 | 1.05E-14 |
period(t) | 0.006673611 | 0.005562379 | 1.199776 | 0.236239 |
x1 | -0.160590278 | 0.466477698 | -0.34426 | 0.732185 |
x2 | 0.136736111 | 0.465780744 | 0.293563 | 0.770384 |
x3 | -0.2439375 | 0.465149266 | -0.52443 | 0.602445 |
x4 | -0.488611111 | 0.464583533 | -1.05172 | 0.298306 |
x5 | -0.363284722 | 0.464083783 | -0.7828 | 0.437671 |
x6 | -0.235958333 | 0.463650232 | -0.50891 | 0.613193 |
x7 | 0.237368056 | 0.463283063 | 0.512361 | 0.610798 |
x8 | -0.517305556 | 0.462982436 | -1.11733 | 0.269531 |
x9 | -0.627979167 | 0.46274848 | -1.35706 | 0.181243 |
x10 | 0.053347222 | 0.462581296 | 0.115325 | 0.908679 |
x11 | -0.187326389 | 0.462480956 | -0.40505 | 0.687281 |
The estimated equation is
Return(y)=4.24975 + 0.00667*period(t) - 0.16059*x1 + 0.13674*x2 -0.24394*x3 -0.48861*x4 - 0.363285*x5 - 0.235958*x6 + 0.237368*x7 - 0.51731*x8 - 0.6279792*x9 + 0.053347*x10 - 0.1873264*x11
The forecasts for first three months of 2011 is as below:
Year | Month | period(t) | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | Forecasted value from estimated equation(yt) |
2011 | Jan | 61 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4.50 |
2011 | Feb | 62 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4.80 |
2011 | Mar | 63 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4.43 |