Economic Models - Convex Optimization
Economic Models - Convex Optimization Economic Models - Convex Optimization
If the rank of ˜ is k, i.e., A Novel Method of Estimation Under Co-Integration 27 r( ˜) = k ≤ n, then f 1 ≥ f 2 ≥···≥f k > 0 and f k+1 = f k+2 =···=f m = 0 F ∗ is diagonal (m × m), and fii ∗ f i ∗ = 1/f i . It should be noted that all the above matrices are real. Also note that if ˜ has full row rank, then ˜ + is the right inverse of ˜, i.e., ˜ ˜ + = I n Hence, the singular value decomposition (SVD) of ˜ is: ˜ = UFV ′ . (10) 4. Computing the Co-Integrating Vectors We proceed to form a matrix F 1 of dimension (m × n) such that: f 1(i,j) = { 0, if i ̸= j √ fi , if i = j (11) In accordance to Eq. (11), we next form a matrix F 2 of dimension (n × m). It is verified that F 1 F 2 = F. Hence, Eq. (10) can be written as: ˜ = AC where A = UF 1 of dimension (n×n) and C = F 2 V of dimension (n×m). After normalization, we get the possible co-integrating vectors as the rows of matrix C, whereas the elements of A can be viewed as approximations to the corresponding coefficients of adjustment. It is apparent that the co-integrating vectors can always be computed, given that r( ˜) > 0. Needless to say that the same steps are followed if , instead of ˜, is considered. 5. Case Study 1: Comparing Co-Integrating Vectors We first consider the VAR(2) model, presented by Holden and Peaman (1994). The x ∼I(1) vector consists of the variables C (consumption), I (income) and W (wealth), so that x ∈ E 3 . We used the same set of data presented at the end of this book (Table D3, pp. 196–198). To simplify the presentation, we did not consider the three dummies DD682, DD792, and DD883. Hence, the VAR in this case has a form analogous to Eq. (1), without trend and dummy components.
28 Alexis Lazaridis Estimating the equations of this VAR by OLS, we get the following results. ⎡ ⎤ ⎡ ⎤ 0.059472 −0.073043 0.0072427 0.064873 ⎢ ⎥ ⎢ ⎥ = ⎣0.5486237 −0.5244858 −0.028041 ⎦ , δ = ⎣ 0.16781451⎦ . 0.3595042 −0.2938160 −0.039400 −0.1554421 (12) Hat (ˆ) is omitted for simplicity. One possible co-integration vector with intercept, obtained by applying the ML method is: 1 − 0.9574698 − 0.048531 + 0.2912925 (13) and corresponds to the long-run relationship: C i = 0.9574698I i + 0.048531W i − 0.2912925. For the vector in Eq. (13) to be a co-integration vector, the necessary and sufficient condition is that the (disequilibrium) errors û i computed from: û i = C i − 0.9574698I i − 0.048531W i + 0.2912925 (14) to be stationary, i.e., {û i }∼I(0), since x ∼I(1). For distinct purposes, these ML errors obtained from Eq. (14) will be denoted by uml i . Applying SVD on ˜ = [.δ] from Eq. (12), we get matrix C which is: ⎡ ⎤ 1 −0.9206225 −0.06545528 0.1110597 ⎢ ⎥ C = ⎣ 1 0.3586208 −0.6305243 −6.403011 ⎦ 1 1.283901 −2.050713 0.4300255 ⎛ ⎞ 1.365344 ⎜ ⎟ Euclidean norm = ⎝ 6.521098 ⎠ . 2.653064 In the column at the right-hand side, the (Euclidean) norm of each row of C is presented. It is noted also that the singular values of are: f 1 = 0.8979247, f 2 = 0.2304381 and f 3 = 0.0083471695. All f i are less than 1, indicating thus that at least one row of C can be regarded as a possible co-integration vector, in the sense that the resulting errors are likely to be stationary. This row usually has the smallest norm,
- Page 2 and 3: ECONOMIC MODELS Methods, Theory and
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- Page 6 and 7: This Volume is Dedicated to the Mem
- 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
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28 Alexis Lazaridis<br />
Estimating the equations of this VAR by OLS, we get the following<br />
results.<br />
⎡<br />
⎤ ⎡<br />
⎤<br />
0.059472 −0.073043 0.0072427<br />
0.064873<br />
⎢<br />
⎥ ⎢<br />
⎥<br />
= ⎣0.5486237 −0.5244858 −0.028041 ⎦ , δ = ⎣ 0.16781451⎦ .<br />
0.3595042 −0.2938160 −0.039400 −0.1554421<br />
(12)<br />
Hat (ˆ) is omitted for simplicity. One possible co-integration vector with<br />
intercept, obtained by applying the ML method is:<br />
1 − 0.9574698 − 0.048531 + 0.2912925 (13)<br />
and corresponds to the long-run relationship:<br />
C i = 0.9574698I i + 0.048531W i − 0.2912925.<br />
For the vector in Eq. (13) to be a co-integration vector, the necessary and<br />
sufficient condition is that the (disequilibrium) errors û i computed from:<br />
û i = C i − 0.9574698I i − 0.048531W i + 0.2912925 (14)<br />
to be stationary, i.e., {û i }∼I(0), since x ∼I(1).<br />
For distinct purposes, these ML errors obtained from Eq. (14) will be<br />
denoted by uml i .<br />
Applying SVD on ˜ = [.δ] from Eq. (12), we get matrix C which is:<br />
⎡<br />
⎤<br />
1 −0.9206225 −0.06545528 0.1110597<br />
⎢<br />
⎥<br />
C = ⎣ 1 0.3586208 −0.6305243 −6.403011 ⎦<br />
1 1.283901 −2.050713 0.4300255<br />
⎛ ⎞<br />
1.365344<br />
⎜ ⎟<br />
Euclidean norm = ⎝ 6.521098 ⎠ .<br />
2.653064<br />
In the column at the right-hand side, the (Euclidean) norm of each row<br />
of C is presented. It is noted also that the singular values of are:<br />
f 1 = 0.8979247, f 2 = 0.2304381 and f 3 = 0.0083471695.<br />
All f i are less than 1, indicating thus that at least one row of C can be<br />
regarded as a possible co-integration vector, in the sense that the resulting<br />
errors are likely to be stationary. This row usually has the smallest norm,
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