TimeSeriesBook.pdf

# One hand that the estimation of φ1 is critical for

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one hand, that the estimation of Φ(1) is critical for the method of moments approach. The IV-approach, on the other hand, depends on the strength or weakness of the instrument used (Pagan and Robertson, 1998; Gospodinov, 2010). It is, of course, possible to combine both short- and long-run restrictions simultaneously. An interesting application of both techniques was presented by Gal´ ı (1992). In doing so, one must be aware that both type of restrictions are consistent with each other and that counting the number of restrictions gives only a necessary condition.

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308 CHAPTER 15. INTERPRETATION OF VAR MODELS Example 3: Identifying Aggregate Demand and Supply Shocks In this example, we follow Blanchard and Quah (1989) and investigate the behavior of the growth rate of real GDP and the unemployment rate for the US over the period from the first quarter 1979 to the second quarter 2004 (102 observations). The AIC and the BIC suggest models of order two and one, respectively. Because some coefficients of b Φ 2 are significant at the 10 percent level, we prefer to use the VAR(2) model which results in the following estimates: 23 b Φ 1 = 0 . 070 - 3 . 376 - 0 . 026 1 . 284 b Φ 2 = 0 . 029 3 . 697 - 0 . 022 - 0 . 320 b Σ = 7 . 074 - 0 . 382 - 0 . 382 0 . 053 . These results can be used to estimate Φ(1) = I 2 - Φ 1 - Φ 2 and consequently also Ψ(1) = Φ(1) - 1 : b Φ(1) = 0 . 901 - 0 . 321 0 . 048 0 . 036 b Ψ(1) = b Φ(1) - 1 = 0 . 755 6 . 718 - 1 . 003 18 . 832 . Assuming that Z t = BV t and following the argument in Section 15.5.1 that the demand shock has no long -run impact on the level of real GDP, we can retrieve an estimate for b 21 : ˆ b 21 = - [ b Ψ(1)] 11 / [ b Ψ(1)] 12 = - 0 . 112 . The solution of the quadratic equation for b 12 are -8.894 and 43.285. As the first solution results in a negative variance for ω 2 2 , we can disregard this solution and stick to the second one. The second solution makes also sense economically, because a positive supply shock leads to positive effects on GDP. Setting b 12 = 43 . 285 gives the following estimates for covariance matrix of the structural shocks Ω: b Ω = b ω 2 d 0 0 b ω 2 s = 4 . 023 0 0 0 . 0016 . 23 The results for the constants are suppressed to save space.
15.5. IDENTIFICATION VIA LONG-RUN RESTRICTIONS 309 The big difference in the variance of both shocks clearly shows the greater importance of demand shocks for business cycle movements. Figure 15.4 shows the impulse response functions of the VAR(2) iden- tified by the long-run restriction. Each figure displays the dynamic effect of a demand and a supply shock on real GDP and the unemployment rate, respectively, where the size of the initial shock corresponds to one standard deviation. The result conforms well with standard economic reasoning. A positive demand shock increases real GDP and lowers the unemployment rate in the short-run. The effect is even amplified for some quarters before it declines monotonically. After 30 quarters the effect of the demand has practically vanished so that its long-run effect becomes zero as imposed by the restriction. The supply shock has a similar short-run effect on real GDP, but initially increases the unemployment rate. Only when the effect on GDP

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