Applied Econometrics assignment3

ECON 3P95
Assignment 3
Milan Doczy , Chenguang Li , Jordan Templeton
June 26, 2016
1

Contents
1 Stationarity and Unit Root Test 4
1.1 Correlogram of SPIndex . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 LB test for SPIndex . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Correlogram for AtlantaHPIndex . . . . . . . . . . . . . . . . . . 6
1.4 Autocorrelation function for AtlantaHPIndex . . . . . . . . . . . 7
1.5 Lag selections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.6 Unit Root Test for SPIndex . . . . . . . . . . . . . . . . . . . . . 9
1.7 Hypothesis test . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.8 Unit Root Test for AtlantaHPIndex . . . . . . . . . . . . . . . . 11
1.9 Hypothesis test . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.10 Time Series Plot AtlantaHPIndex & SPIndex . . . . . . . . . . . 13
2 Cointegrating Regression 14
2.1 Scatterplot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Estimations of the Cointegration Residual . . . . . . . . . . . . . 15
2.3 Correlogram of the Cointegration Residual . . . . . . . . . . . . . 16
2.4 Autocorrelation function of cointegraion Residuals . . . . . . . . 17
3 Engle-Granger Test of Cointegration 18
3.1 Hypothesis test . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4 Estimation of the VAR Model with the Optimal Lag Choice 20
4.1 Output for Optimal Lag Choice for VAR . . . . . . . . . . . . . . 20
5 VAR Estimation continue 21
5.1 The Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
5.2 Visual Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5.2.1 Combined time series SPIndex VAR Atalanta VAR . . . 22
5.2.2 Correlogram . . . . . . . . . . . . . . . . . . . . . . . . . 23
5.3 White Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5.3.1 LB test for SPIndex . . . . . . . . . . . . . . . . . . . . . 25
5.3.2 LB test for AtalantaHPIndex . . . . . . . . . . . . . . . . 26
6 In-sample VAR forecasts 27
6.1 In-sample SPIndex VAR forecasts . . . . . . . . . . . . . . . . . . 27
6.2 In-sample AtlantaHPIndex VAR forecasts . . . . . . . . . . . . . 28
7 AR(1) in-sample forecasts 29
7.1 AR(1) for SPIndex . . . . . . . . . . . . . . . . . . . . . . . . . . 29
7.2 AR(1) for AtlantaHPIndex . . . . . . . . . . . . . . . . . . . . . 30
8 Out-of sample VAR forecasting 31
8.1 Out-of sample forecasting for SPIndex VAR . . . . . . . . . . . . 31
8.2 Out-of sample forecasting for AtlantaHPIndex . . . . . . . . . . . 32
2

9 Appendix 33
9.1 Leg Slection OLS regression for unit root test . . . . . . . . . . . 33
9.1.1 Gretl code for OLS regression (SPIndex 10 legs) . . . . . 33
9.1.2 Gretl code for OLS regression (AtlantaHPIndex 10 legs) . 34
9.2 cointegrating regression . . . . . . . . . . . . . . . . . . . . . . . 35
9.2.1 Gretl code . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
9.3 VAR in sample forecasting . . . . . . . . . . . . . . . . . . . . . . 36
9.3.1 In-sample VAR output . . . . . . . . . . . . . . . . . . . . 36
9.3.2 AR(1) In-sample forescast . . . . . . . . . . . . . . . . . . 38
9.4 Out-of sample VAR forecasting . . . . . . . . . . . . . . . . . . . 39
9.4.1 Out put of Out-of sample VAR forecasting . . . . . . . . 39
9.5 Gretl: command log . . . . . . . . . . . . . . . . . . . . . . . . . 41
3

1 Stationarity and Unit Root Test
1.1 Correlogram of SPIndex
Figure 1: correlagram of S&P 500 Index
From the correlogram of SPIndex we can see ACF starts high and is falling
through time but is still pretty high after 20 lags, which indicates a nonstation-
ary data set.
4

1.2 LB test for SPIndex
LAG ACF PACF Q-STAT. [p-value]
1 0.9748 *** 0.0101629 *** -1.8636 0.0641
2 0.452030 *** 0.0702484 6.4347 0.0000
3 0.9241 *** -0.0130 481.6398 0.0000
4 0.8990 *** -0.0095 627.2279 0.0000
5 0.8746 *** 0.0014 465.8330 0.0000
6 0.8508 *** -0.0010 897.7754 0.0000
7 0.8275 *** -0.0030 1023.3324 0.0000
8 0.8044 *** -0.0082 1142.6946 0.0000
9 0.7814 *** -0.0101 1256.0196 0.0000
10 0.4585 *** -0.0112 1363.4503 0.0000
11 0.7357 *** -0.0103 1465.1438 0.0000
12 0.7131 *** -0.0035 1652.0594 0.0000
13 0.6909 *** -0.0035 1737.8513 0.0000
14 0.6695 *** 0.0026 1737.8513 0.0000
15 0.6488 *** 0.0009 1818.9225 0.0000
16 0.6287 *** 0.0006 1895.5384 0.0000
17 0.6094 *** 0.0022 1967.9673 0.0000
18 0.5904 *** -0.0033 2036.3975 0.0000
19 0.5904 *** -0.0033 2101.0093 0.0000
20 0.5538 *** -0.0013 2261.9961 0.0000
21 0.5362 *** -0.0018 2219.5375 0.0000
22 0.5189 *** -0.0032 2273.7922 0.0000
Furthermore, non-stationarity is proven by the LB test that all the P values
are near 0. Indecates there is no white noise and still some dynamics for us to
capture. Therefore we need to use proper unit root test to determine whether
our data is stationary.
5

1.3 Correlogram for AtlantaHPIndex
Figure 2: correlagram of Atlanta House Pricing Index
From the correlogram of speciﬁc AtlantaHPIndex we can see ACF is classic
“start high and stay high”, which indicates a nonstationary data set.
6

1.4 Autocorrelation function for AtlantaHPIndex
1 0.9842 *** 0.9842 *** 171.4817 0.0000
2 0.9687 *** -0.0022 338.5430 0.0000
3 0.9553 *** -0.0005 501.2976 0.0000
4 0.9385 *** 0.0096 659.9632 0.0000
5 0.9238 *** -0.0052 814.5936 0.0000
6 0.9091 *** -0.0044 965.2503 0.0000
7 0.8943 *** -0.0115 111.9217 0.0000
8 0.8795 *** -0.0079 1254.6383 0.0000
9 0.8644 *** -0.0204 1393.3069 0.0000
10 0.8487 *** -0.0229 1527.8208 0.0000
11 0.8329 *** -0.0147 1658.1635 0.0000
12 0.8170 *** -0.0140 1784.3336 0.0000
13 0.8010 *** -0.0083 1906.3875 0.0000
14 0.7850 *** -0.0141 2024.3255 0.0000
15 0.7690 *** -0.0058 2138.2272 0.0000
16 0.7533 *** -0.0001 2248.2213 0.0000
17 0.7377 *** -0.0053 2354.3853 0.0000
18 0.7220 *** -0.0135 2456.7193 0.0000
19 0.7065 *** -0.0015 2555.3333 0.0000
20 0.6907 *** -0.0179 2650.1943 0.0000
21 0.6746 *** -0.0170 2741.2887 0.0000
22 0.6580 *** -0.0278 2828.5197 0.0000
Furthermore, non-stationarity is proven by the LB test that all the P values
are near 0. Indecates there is no white noise and still some dynamics for us to
capture. Therefore we need to use proper unit root test to determine whether
our data is stationary.
7

1.5 Lag selections
Table 1: BIC results for diﬀerent lag of SPIndex & AtlantaHPIndex
Lag S&P Atlanta
1 -1725.376* -1539.490
2 -1718.734 -1527.382
3 -1719.351 -1547.213*
4 -1714.473 -1540.906
5 -1700.517 -1526.633
6 -1685.497 -1524.254
7 -1669.186 -1512.989
8 -1656.754 -1502.498
9 -1644.221 -1489.383
10 -1631.732 -1490.806
From the unit root lag selections we choose the lowest BIC. Therefore it is
ﬁrst lag for S&PIndex and third lag for AtlantaHPIndex
8

.
1.6 Unit Root Test for SPIndex
Table 2: Model 24: OLS, using observations 1991:03–2005:06 (T = 172)
Dependent variable: d l SPIndex
Coeﬃcient Std. Error t-ratio p-value
const 0.000626609 0.00392999 0.1594 0.8735
time 1.53504e-005 7.1412e-006 2.5675 0.0111
l SPIndex 1 -0.000249928 0.000952091 -0.2625 0.7933
d l SPIndex 1 0.820463 0.044563 18.4195 0.0000
Mean dependent var 0.005683 S.D. dependent var 0.005705
Sum squared resid 0.000393 S.E. of regression 0.001530
R2
0.929369 Adjusted R2
0.928108
F(3,168) 736.8531 P-value(F) 2.07e-96
Log-likelihood 872.9830 Akaike criterion -1737.966
Schwarz criterion -1725.376 H-Q -1732.858
ˆp 0.162052 Durbin’s h 2.618554
In order to preform a proper unit root test we have to select the lag that has
the lowest SIC value as our optimal lag to model. After generating the model
in gretl (Appendix 9.3). From the output (Appendix 9.3.2) of our model we
choose lag one as our unit root test optimal lag. The results of the unit root
test are shown above. We can see the t-ratio of the log of the SPIndex value is
less than the absolute value of 3.9, this indicates non-stationarity for SPIndex.
9

1.7 Hypothesis test
H0: dlSPIndex = 1 → UnitRoot; non − stationary
H1: dlSPIndex < 1 → NoUnitRoot; stationary
∆SPIndex = β ∗ 0 + β1 ∗ (SPIndex)t−1 + ∆SPIndext−1 + +ut
10

1.8 Unit Root Test for AtlantaHPIndex
Model 1: OLS, using observations 1991:05–2005:06 (T = 170)
Dependent variable: d l AtlantaHPIndex
Coeﬃcient Std. Error t-raio p-value
const 0.102789 0.0356068 2.8868 0.0044
l AtlantaHPIndex 1 -0.0242917 0.00853775 -2.8452 0.0050
time 0.000102817 3.35030e-005 3.0689 0.0025
d l AtlantaHPIndex 1 0.446385 0.0692713 6.4440 0.0000
d l AtlantaHPIndex 3 -0.395986 0.0682175 -5.8048 0.0000
R2
0.102183
F(5,164) 23.73902 P-value(F) 6.48e-18
ˆp 0.088891 Durbin’s h 2.700071
Same procedures for AtlantaHPIndex. After generating the model in gretl (Ap-
pendix 9.4). From the output (Appendix 9.4.2) of our model we choose lag three
as our unit root test optimal lag. The results of the unit root test are shown
above. We can see the t-ratio of the log of the AtlantaHPIndex value is less than
the absolute value of 3.9, this indicates non-stationary for AtlantaHPIndex.
11

1.9 Hypothesis test
H0: dlAtlantaHPIndex = 1 → UnitRoot; non − stationary
H1: dlAtlantaHPIndex < 1 → NoUnitRoot; stationary
∆AtlantaHPIndex = β∗0+β1∗(AtlantaHPIndex)t−1+∆AtlantaHPIndext−1+
∆AtlantaHPIndext−2 + ∆AtlantaHPIndext−3 + ut
12

1.10 Time Series Plot AtlantaHPIndex & SPIndex
Figure 3: Time series Plot AtlantaHPIndex & SPIndex
From the time series of both SPIndex and Atlanta houseing price index we
can see some correlations between the two. It seems both index are moving
towards a increasing trend. However For SPIndex appears a more qudratic
trend and Atlanta Index is more linear compare to SPIndex.
13

2 Cointegrating Regression
2.1 Scatterplot
Figure 4: scatterplot AtlantaHPIndex & SPIndex (with least squares ﬁt)
The scatterplot shows that the Atlanta index moves with the ten city index.
The best ﬁt line follows the general trend of the data and has a positive slope.
This means that according to our graph when x increases, y increases.
14

2.2 Estimations of the Cointegration Residual
Lag BIC
1 -1548.349
2 -1537.330
3 -1553.980*
4 -1548.994
5 -1534.946
6 -1531.406
7 -1520.662
8 -1509.729
9 -1496.274
10 -1498.845
In order to preform a proper cointegration regression we have to select the
lag that has the lowest SIC value as our optimal lag to model. After generating
the model in gretl . From the output of our model we choose lag three as our
engle-granger test optimal lag.
15

2.3 Correlogram of the Cointegration Residual
Figure 5: Correlogram of the cointgration residuals
ACF is falling over time and falls into the conﬁdence interval and PACF
jumped to 0 after lag 1, which indicates stationary data.
16

2.4 Autocorrelation function of cointegraion Residuals
1 0.9721 *** 0.9721 *** 167.2790 0.0641
2 0.9358 *** -0.1664 ** 323.2093 0.0000
3 0.8927 *** -0.1243 465.9264 0.0000
4 0.8495 *** 0.0078 595.9203 0.0000
5 0.8059 *** -0.0220 713.6020 0.0000
6 0.7632 *** -0.0101 819.7695 0.0000
7 0.7259 *** 0.0765 916.4095 0.0000
8 0.6912 *** -0.0050 1004.5484 0.0000
9 0.6614 *** 0.0428 1085.7421 0.0000
10 0.6343 *** 0.0078 1160.8656 0.0000
11 0.6055 *** -0.0776 1229.7398 0.0000
12 0.5709 *** -0.1218 1291.3620 0.0000
13 0.5245 *** -0.2084 1343.6957 0.0000
14 0.4711 *** -0.0994 1386.1673 0.0000
15 0.4130 *** -0.0528 1419.0153 0.0000
16 0.3565 *** 0.0305 1443.6479 0.0000
17 0.3038 *** 0.0448 1461.6534 0.0000
18 0.2570 *** 0.0492 1474.6184 0.0000
19 0.2098 *** -0.1170 1483.3126 0.0000
20 0.1661 ** -0.0256 1488.8020 0.0000
21 0.1272 * -0.0036 1492.0410 0.0000
22 0.0978 0.1064 1493.9692 0.0000
However according to the LB test all the P values are near 0 which indicates
non-stationary data. This is beaucase gretl is interpreting the LB test results
strictly. Therefore we can say that the cointegration residuals are weakly sta-
tionary.
17

3 Engle-Granger Test of Cointegration
Dependent variable: d CointRe
Coeﬃcient Std. Error t-raio p-value
const -0.00019022 0.0001854 -1.026 0.3065
Coint 1 -0.0189401 0.0101629 -1.855 0.0641
d CointRe 1 0.452030 0.0702484 6.4347 0.0000
d CointRe 2 0.331079 0.0753387 4.3945 0.0000
d CointRe 3 -0.369857 0.0698553 -5.2946 0.0000
Mean dependent var -0.000265 S.D. dependent var 0.002976
R2
0.362288
F(4,166) 24.75252 P-value(F) 4.40e-16
ˆp 0.099597 Durbin’s h 3.235601
The results of the Engle-Granger test are shown above. We can see that the t-
ratio of the log of the cointegration residuals’ value is less than the absolute value
of 3.9, which indicates the residuals from the cointegration are not stationary.
Since the residuals are nonstationary. We need to estimate a VAR instead of a
VECM and lag selection.
18

3.1 Hypothesis test
H0: CointRe1 = 1 → UnitRoot; non − stationary
H1: CointRe1 < 1 → NoUnitRoot; stationary
∆CointRe1 = CointRe1 + CointRe1t + CointRe1t−1 + ut
19

4 Estimation of the VAR Model with the Opti-
mal Lag Choice
4.1 Output for Optimal Lag Choice for VAR
Lag BIC
1 -19.144385*
2 -19.023342
3 -19.106130
4 -19.063737
Useing gretl VAR lag slection to choose optimal lag choices for VAR. The lowest
BIC is lag 1 so that will be our optimal lag choice for VAR and the results are
shown below.
20

5 VAR Estimation continue
5.1 The Fit
In equation 1, the adjusted R2
is 0.923847 so 92.3847% of the variation in the
model is explained by the independent variables. Which is a very strong fit.
The SER is 0.001574 so the average distance between the actual data and our
fitted values is 0.1574% which is relatively small. Looking at the p-values we
can see that the lag of the composite index is significant, but the lag of the
Atlanta index is not. Overall equation 1 has a good fit.
For equation 2 the adjusted R2
is 0.246231 so only 24.6231% of the variation
in the model is explained. The SER is 0.002653 so the average distance between
actual data and our fitted value is 0.2653% much larger than equation 1. Finally
looking at the p-values we can see that the lag of the Atlanta index is significant
while the lag of the composite index is not at the 90% confidence level. Overall
the fit of equation 2 is not very strong.
21

5.2 Visual Diagnostics
5.2.1 Combined time series SPIndex VAR Atalanta VAR
Figure 6: Combined time series for SPIndex VAR & AtlantaHPIndex VAR
22

5.2.2 Correlogram
Figure 7: Correlogram of S&P VAR residuals
ACF are pretty much within conﬁdence interval and only a little bit out of
it. Therefore we would say SPIndex are stationary.
23

Figure 8: Correlogram of AtlantaHPIndex VAR Residual
There are some ACF values are out of the conﬁdence interval but it is not
too much to say it is non-stationary. Therefore we would say AtlantaHPIndex
are weakly stationary.
24

5.3 White Noise
5.3.1 LB test for SPIndex
1 0.1097 0.1097 2.1074 0.147
2 0.1275 0.1142 4.8502 0.088
3 -0.1195 -0.1478 * 7.3770 0.061
4 -0.1824 ** -0.1768 ** 13.3037 0.010
5 -0.1675 ** -0.1061 18.3308 0.003
6 -0.0913 -0.0387 19.8324 0.003
7 0.0880 0.1015 21.2367 0.003
8 0.0956 0.0431 22.9043 0.003
9 0.1542 ** 0.0654 27.2687 0.001
10 0.0189 *** -0.0394 27.3350 0.002
11 -0.1113 *** -0.1283 * 29.6391 0.002
12 -0.1640 ** -0.0973 34.6680 0.001
13 -0.1385 * -0.0354 38.2770 0.000
14 -0.0434 0.0114 38.6340 0.000
15 0.0650 0.0472 39.4390 0.001
16 0.1371 * 0.0425 43.0449 0.000
17 0.0440 -0.0751 43.4190 0.000
18 0.1735 ** 0.1373 * 49.2684 0.000
19 -0.0594 -0.0355 49.9592 0.000
20 0.0723 0.1359 * 50.9878 0.000
21 -0.0440 0.0450 51.3721 0.000
22 -0.0030 0.0062 51.3739 0.000
For SPIndex VAR, the P values of LB tests are pretty much all 0. This
indecates that SPIndex VAR is non-stationary. However from the correlogram
we know that gretl is being strict for the ACF that are out of the coﬁdence
interval. Therefore we could call SPIndex VAR is weakly stationary.
25

5.3.2 LB test for AtalantaHPIndex
1 -0.0583 -0.0583 0.5949 0.441
2 0.3143 *** 0.3120 *** 17.9880 0.000
3 -0.3621 *** -0.3683 *** 41.2057 0.000
4 0.1474 * 0.0766 4530768 0.000
5 -0.1310 * 0.0974 48.1499 0.000
6 -0.0060 -0.2608 *** 48.1564 0.000
7 -0.1571 ** -0.0548 52.6352 0.000
8 -0.0112 0.0734 52.6583 0.000
9 -0.0758 -0.1561 53.7125 0.000
10 0.0620 0.0395 ** 54.4235 0.000
11 0.1589 ** 0.3290 *** 59.1193 0.000
12 0.2521 *** 0.1488 * 71.0115 0.000
13 0.2439 *** 0.1653 ** 82.2055 0.000
14 0.0763 0.1605 ** 83.3123 0.000
15 -0.0040 -0.0570 83.3123 0.001
16 -0.1467 * -0.1614 ** 87.4399 0.000
17 -0.1007 -0.0479 89.3993 0.000
18 -0.0734 0.0325 90.4468 0.000
19 -0.0033 0.0158 90.4489 0.000
20 -0.1125 -0.1011 92.9404 0.000
21 -0.0708 -0.0365 93.9351 0.000
22 -0.0903 -0.0950 95.5607 0.000
The situation is the same for AtlantaHPIndex VAR. From its LB tests all the
P values are near 0, indicates non-stationary. However if we look at the correl-
ogram we can see almost all the ACF are within conﬁdence interval. Therefore
we could call it weakly stationary as well.
26

6 In-sample VAR forecasts
6.1 In-sample SPIndex VAR forecasts
Figure 9: In-sample VAR Forescast for SPIndex
Mean Squared Error 0.00015244
Mean Absolute Error 0.0093385
Looking at the above graph we can see that our forecast drastically over-
estimate the SP Index. Clearly our model was unable to predict the ﬁnancial
meltdown of 2007-2008. Once the market recovers by 2010 our forecast looks
better, but it still under and over predicts the SP Index. Our mean squared
error is only 0.00015244 and since the MSE is the average distance between the
actual values and forecasted values in square terms so our forecasting method
is accurate.
27

6.2 In-sample AtlantaHPIndex VAR forecasts
Figure 10: In-sample VAR Forescast for AtlantaHPIndex
Looking at the above graph we can see that our forecast is very diﬀerent from
the actual values, however it does seem like our forecast accurately predicts the
average trend of the Atlanta index. We can also see some similarities between
Atlanta and the composite data. When Atlanta is mostly below zero so is the
composite, likewise when Atlanta is mostly above zero so is the composite. The
MSE of this forecast 0.00027322 is a little larger than our last MSE, but still
relatively small. Therefore Our forecast for Atlanta is not as accurate as our
forecast for the composite data.
28

7 AR(1) in-sample forecasts
7.1 AR(1) for SPIndex
Figure 11: AR(1) In-sample forescast for SPIndex
Looking at our forecast using AR(1) it’s clear that our predictions vastly
overestimate the actual values of the composite data. This is conﬁrmed by the
fact that the MSE of this forecast is 0.00047428 which is over three times larger
than our MSE using the VAR approach. Therefore the VAR approach is more
accurate.
29

7.2 AR(1) for AtlantaHPIndex
Figure 12: AR(1) In-sample forescast for AtlantaHPIndex
Just like with the VAR approach we can see that our forecast regularly under
and over predicts the actual values for Atlanta, but does seem to capture the
average eﬀectively. The MSE under this method is 0.00028477 which is only
slightly larger than 0.00027322 the MSE under the VAR approach. Therefore
we ﬁnd the VAR approach to be more accurate, but only barely.
30

8 Out-of sample VAR forecasting
8.1 Out-of sample forecasting for SPIndex VAR
Figure 13: Out-of sample forecasting for SPIndex VAR
Our out of sample forecast for the composite data shows a gradual decline in
housing prices. Declining housing prices can lead to more consumers defaulting
on their morgages so banks will lose money. It will also lead to consumers having
less wealth which can lead to lower spending. Lower spending can decrease the
economic prosperity of the United States which can lead to a recession.
31

8.2 Out-of sample forecasting for AtlantaHPIndex
Figure 14: Out-of sample forecasting for AtlantaHPIndex
Our out of sample forecast for Atlanta show a very gradual increase in hous-
ing prices. Increasing housing prices increase the wealth of home owners which
can lead to higher consumption. This will help to encourage economic growth
in the United States.
32

9 Appendix
9.1 Leg Slection OLS regression for unit root test
9.1.1 Gretl code for OLS regression (SPIndex 10 legs)
ols d_l_SPIndex 0 time l_SPIndex(-1) d_l_SPIndex(-1)
ols d_l_SPIndex 0 time l_SPIndex(-1) d_l_SPIndex(-1 to -2)
33

9.1.2 Gretl code for OLS regression (AtlantaHPIndex 10 legs)
ols d_l_AtlantaHPIndex time l_AtlantaHPIndex(-1) d_l_AtlantaHPIndex(-1)
ols d_l_AtlantaHPIndex time l_AtlantaHPIndex(-1) d_l_AtlantaHPIndex(-1 to -2)
34

9.2 cointegrating regression
9.2.1 Gretl code
ols d_CointRe CointRe(-1) d_CointRe(-1)
ols d_CointRe CointRe(-1) d_CointRe(-1 to -2)
35

9.3 VAR in sample forecasting
9.3.1 In-sample VAR output
VAR system, lag order 1
OLS estimates, observations 1991:03–2005:06 (T = 172)
Log-likelihood = 1645.37
Determinant of covariance matrix = 1.68288e–011
AIC = −19.0624
BIC = −18.9526
HQC = −19.0179
Portmanteau test: LB(43) = 290.434, df = 168 [0.0000]
Equation 1: d l SPIndex
const 0.000362622 0.000200586 1.8078 0.0724
d l SPIndex 1 0.949869 0.0215352 44.1077 0.0000
R2
0.923847
F(2,169) 1038.238 P-value(F) 1.18e95
ˆp 0.109826 Durbin-Watson 1.770112
F-tests of zero restrictions
All lags of dlSPIndex F(1, 169) = 1945.49 [0.0000]
All lags of dlAtlantaHPIndex F(1, 169) = 0.0139034 [0.9063]
36

Equation 2: d l AtlantaHPIndex
const 0.00166369 0.000337961 4.9227 0.000
d l SPIndex 1 0.0589337 0.0362840 1.6242 0.1062
R2
0.246231
F(2,169) 28.92997 P-value(F) 1.57e-11
ˆp -0.059246 Durbin-Watson 2.100654
37

9.3.2 AR(1) In-sample forescast
Dependent variable: d l SPIndex
const -0.000402302 0.000284916 -1.4120 0.1598
time 1.70494e-005 5.13528e-006 3.3200 0.0011
d l SPIndex 1 0.819612 0.0443024 18.5004 0.0000
R2
0.246231
F(2,169) 28.92997 P-value(F) 1.57e-11
Schwarz criterion -1539.652 Hannan-Quinn -1545.263
Dependent variable: d l AtlantaHPIndex
const 0.00142508 0.000444203 3.2082 0.0016
time 6.10844e-006 1.06303e-006 1.4673 0.1442
d l SPIndex 1 0.467432 0.0669743 6.9793 0.0000
R2
0.244094
F(2,169) 28.60935 P-value(F) 1.99e{11
Schwarz criterion -1539.652 Hannan-Quinn -1545.263
ˆp -0.064760 Durbin-Watson -1.776816
38

9.4 Out-of sample VAR forecasting
9.4.1 Out put of Out-of sample VAR forecasting
VAR system, lag order 1
OLS estimates, observations 1991:03–2015:12 (T = 298)
Log-likelihood = 2482.07
Determinant of covariance matrix = 1.99754e–010
AIC = −16.6179
BIC = −16.5435
HQC = −16.5881
Portmanteau test: LB(48) = 899.196, df = 188 [0.0000]
Equation 1: d l SPIndex
const 0.000178243 0.000139241 1.2801 0.2015
d l SPIndex 1 0.957289 0.0181784 52.6607 0.0000
d l AtlantaHPIndex 1 -0.000291567 0.0132617 -0.0220 0.9825
R2
0.919568
F(2,169) 1698.775 P-value(F) 1.3e162
39

Equation 2: d l AtlantaHPIndex
const 8.88034e-005 0.000399593 0.2222 0.8243
d l SPIndex 1 0.133320 0.0521682 2.5556 0.0111
R2
0.648333
F(2,169) 274.7739 P-value(F) 4.18e-68
40

9.5 Gretl: command log
Question 1
rename 1 SPIndex
logs SPIndex
logs AtlantaHPIndex
diff AtlantaHPIndex
diff SPIndex
genr time
smpl 1991:01 2005:06
gnuplot SPIndex AtlantaHPIndex --time-series --with-lines
Question 2
corrgm l_SPIndex 22
corrgm l_AtlantaHPIndex 22
# model 1
ols d_SPIndex 0 time l_SPIndex(-1) d_SPIndex(-1)
delete d_AtlantaHPIndex d_SPIndex
diff l_SPIndex
diff l_AtlantaHPIndex
delete d_SPIndex_1
# model 2
ols d_l_SPIndex 0 l_SPIndex(-1) time d_l_SPIndex(-1)
# model 3
ols d_l_AtlantaHPIndex 0 time l_AtlantaHPIndex(-1) d_l_AtlantaHPIndex(-1 _to -3)
Question 3
gnuplot l_AtlantaHPIndexx l_SPIndex
Question 4
ols l_SPIndex 0 l_AtlantaHPIndex(-1)
series CointRe = $CointRe$
gnuplot CointRe --time-series --with-lines
corrgm CointRe 22
Question 5
ols d_CointRe CointRe(-1) d_CointRe(-1)
41

Question 6
freq uhat11 --normal
corrgm uhat12 22
freq uhat12 --normal
var 3 d_l_SPIndex d_l_HomePriceIndex
smpl 1991:01 2015:12
var 3 d_l_SPIndex d_l_HomePriceIndex --lagselect
Question 7
smpl 1991:01 2015:12
diff l_SPIndex
diff l_HomePriceIndex
smpl 1991:01 2005:06
Question 9
smpl 1991:01 2015:12
smpl --full
diff l_HomePriceIndex
smpl 1991:01 2015:12
smpl 1991:01 2005:06
smpl --full
42

References
[1] The History of Recessions in the United States
http://useconomy.about.com/od/grossdomesticproduct/a/recession histo.htm
[2] A publication of the Board of Governors of the Federal Reserve System
http://www.federalreserve.gov/pf/pf.htm
43

Applied Econometrics assignment3

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