Economic Models - Convex Optimization
Economic Models - Convex Optimization Economic Models - Convex Optimization
A Novel Method of Estimation Under Co-Integration 35 The series { ∆xi}, { ∆ 2 x i }, and { ∆ 3 x i } Figure 2. Differences of the NI series {x i }. Using the series {x i } as shown in Table 1, we can easily verify from the corresponding graph, that even for n = 5, the series { 5 x i } is not stationary. In Fig. 2, the series {x i }, { 2 x i }, and { 3 x i } are presented, for a better understanding of this situation. We will call the variables, which belong to this category near the integrated (NI) (Banerjee et al., 1993, p. 95) series. To trace that a given series is NI, we may run the regression: x i = β 1 + β 2 x i−1 + β 3 t i + q∑ β j+3 x i−j + u i (24) which is a re-parametrized AR(q) with constant, where we added a trend too. As shown in Eq. (17), the value of q is set such that the noises u i to be white. A comparatively simple way for this verification is to compute the residuals û i and to consider the corresponding Ljung-Box Q statistics and particularly their p-values, which should be much greater than 0.1, to say that no autocorrelation (AC) is present. On the other hand, if we trace heteroscedasticity, this is a strong indication that we have a NI series. For the data presented in Table 1, we found that q = 2. The corresponding Q statistics (Column 4) together with p-values are presented in Table 2. We see that for all k (column 1), the corresponding p-values (column 5) are greater than 0.1. But, if we look at the residuals graph (Fig. 3), we can verify the presence of heteroscedasticity. j=1
36 Alexis Lazaridis Table 2. Residuals: Autocorrelations, PAC, Q statistics and p-values. Autocorrelation (AC), partial autocor coefficients (PAC) Q statistics and p values (prob). Values AC PAC Q Stat(L-B) p value Q Stat(B-P) p value of k 1 0.040101 0.040101 0.072482 0.787756 0.067540 0.794952 2 −0.061329 −0.063039 0.246254 0.884151 0.225515 0.893367 3 −0.008189 −0.003064 0.249432 0.969240 0.228331 0.972891 4 −0.285494 −0.290504 4.213238 0.377916 3.651618 0.455202 5 0.042856 0.072784 4.304969 0.506394 3.728756 0.589090 6 0.099724 0.058378 4.815469 0.567689 4.146438 0.656867 7 −0.151904 −0.167680 6.033816 0.535806 5.115577 0.645861 8 −0.007846 −0.069642 6.037162 0.643069 5.118163 0.744875 9 0.001750 0.023203 6.037333 0.736176 5.118291 0.823877 10 0.109773 0.165620 6.733228 0.750367 5.624397 0.845771 11 0.134538 0.022368 7.812250 0.730017 6.384616 0.846509 12 −0.029101 −0.048363 7.864417 0.795634 6.420185 0.893438 13 −0.074997 −0.028063 8.222832 0.828786 6.656414 0.918979 14 −0.084284 −0.017969 8.691681 0.850280 6.954772 0.936441 15 −0.113984 −0.104484 9.580937 0.845240 7.500452 0.942248 16 −0.081662 −0.157803 10.05493 0.863744 7.780540 0.955133 17 −0.001015 −0.005026 10.05501 0.901288 7.780582 0.971016 18 −0.053178 −0.056574 10.27276 0.922639 7.899356 0.980097 Figure 3. The residuals from Eq. (24) (q = 2).
- Page 8 and 9: Contents Tom Oskar Martin Kronsjo:
- Page 10 and 11: Tom Oskar Martin Kronsjo: A Profile
- Page 12 and 13: Tom Oskar Martin Kronsjo students t
- Page 14 and 15: About the Editor Prof. Dipak Basu i
- Page 16 and 17: Contributors Athanasios Athanasenas
- Page 18: Contributors Victoria Miroshnik, is
- Page 21 and 22: xx Introduction type of model is ve
- Page 23 and 24: This page intentionally left blank
- Page 25 and 26: This page intentionally left blank
- Page 27 and 28: 4 Olav Bjerkholt Among these was al
- Page 29 and 30: 6 Olav Bjerkholt method, searching
- Page 31 and 32: 8 Olav Bjerkholt context, but one o
- Page 33 and 34: 10 Olav Bjerkholt be modified in th
- Page 35 and 36: 12 Olav Bjerkholt through a number
- Page 37 and 38: 14 Olav Bjerkholt memoranda. These
- Page 39 and 40: 16 Olav Bjerkholt hopelessly ineffe
- Page 41 and 42: 18 Olav Bjerkholt References Arrow,
- Page 43 and 44: This page intentionally left blank
- Page 45 and 46: This page intentionally left blank
- Page 47 and 48: 24 Alexis Lazaridis A straightforwa
- Page 49 and 50: 26 Alexis Lazaridis where ⎛ ⎞ j
- Page 51 and 52: 28 Alexis Lazaridis Estimating the
- Page 53 and 54: 30 Alexis Lazaridis 5.1. Testing fo
- Page 55 and 56: 32 Alexis Lazaridis by applying the
- Page 57: 34 Alexis Lazaridis Nevertheless, t
- Page 61 and 62: 38 Alexis Lazaridis Table 3. Critic
- Page 63 and 64: 40 Alexis Lazaridis Figure 4. The s
- Page 65 and 66: 42 Alexis Lazaridis Perron, P and T
- Page 67 and 68: 44 Dipak R. Basu and Alexis Lazarid
- Page 69 and 70: 46 Dipak R. Basu and Alexis Lazarid
- Page 71 and 72: 48 Dipak R. Basu and Alexis Lazarid
- Page 73 and 74: 50 Dipak R. Basu and Alexis Lazarid
- Page 75 and 76: 52 Dipak R. Basu and Alexis Lazarid
- Page 77 and 78: 54 Dipak R. Basu and Alexis Lazarid
- Page 79 and 80: 56 Dipak R. Basu and Alexis Lazarid
- Page 81 and 82: 58 Dipak R. Basu and Alexis Lazarid
- Page 83 and 84: 60 Dipak R. Basu and Alexis Lazarid
- Page 85 and 86: 62 Dipak R. Basu and Alexis Lazarid
- Page 87 and 88: 64 Dipak R. Basu and Alexis Lazarid
- Page 89 and 90: 66 Dipak R. Basu and Alexis Lazarid
- Page 91 and 92: This page intentionally left blank
- Page 93 and 94: 70 Andrew Hughes Hallet have, there
- Page 95 and 96: 72 Andrew Hughes Hallet other. The
- Page 97 and 98: 74 Andrew Hughes Hallet success in
- Page 99 and 100: 76 Andrew Hughes Hallet with a hori
- Page 101 and 102: 78 Andrew Hughes Hallet control inf
- Page 103 and 104: 80 Andrew Hughes Hallet Table 2. Fi
- Page 105 and 106: 82 Andrew Hughes Hallet country 13
- Page 107 and 108: 84 Andrew Hughes Hallet Given this
36 Alexis Lazaridis<br />
Table 2.<br />
Residuals: Autocorrelations, PAC, Q statistics and p-values.<br />
Autocorrelation (AC), partial autocor coefficients (PAC) Q statistics and p values (prob).<br />
Values AC PAC Q Stat(L-B) p value Q Stat(B-P) p value<br />
of k<br />
1 0.040101 0.040101 0.072482 0.787756 0.067540 0.794952<br />
2 −0.061329 −0.063039 0.246254 0.884151 0.225515 0.893367<br />
3 −0.008189 −0.003064 0.249432 0.969240 0.228331 0.972891<br />
4 −0.285494 −0.290504 4.213238 0.377916 3.651618 0.455202<br />
5 0.042856 0.072784 4.304969 0.506394 3.728756 0.589090<br />
6 0.099724 0.058378 4.815469 0.567689 4.146438 0.656867<br />
7 −0.151904 −0.167680 6.033816 0.535806 5.115577 0.645861<br />
8 −0.007846 −0.069642 6.037162 0.643069 5.118163 0.744875<br />
9 0.001750 0.023203 6.037333 0.736176 5.118291 0.823877<br />
10 0.109773 0.165620 6.733228 0.750367 5.624397 0.845771<br />
11 0.134538 0.022368 7.812250 0.730017 6.384616 0.846509<br />
12 −0.029101 −0.048363 7.864417 0.795634 6.420185 0.893438<br />
13 −0.074997 −0.028063 8.222832 0.828786 6.656414 0.918979<br />
14 −0.084284 −0.017969 8.691681 0.850280 6.954772 0.936441<br />
15 −0.113984 −0.104484 9.580937 0.845240 7.500452 0.942248<br />
16 −0.081662 −0.157803 10.05493 0.863744 7.780540 0.955133<br />
17 −0.001015 −0.005026 10.05501 0.901288 7.780582 0.971016<br />
18 −0.053178 −0.056574 10.27276 0.922639 7.899356 0.980097<br />
Figure 3. The residuals from Eq. (24) (q = 2).
Change language